The fractional loss of kinetic energy = (Ei - Ef)/Ei = M/(m+M)
How to determine fractional loss of kinetic energy?To calculate the ratio M/(m+M), we need to add up the values of m and M in Data Table 1:
m = 0.023 kg
M = 0.372 kg
m+M = 0.023 + 0.372 = 0.395 kg
M/(m+M) = 0.372/0.395 = 0.942
In Question 4, we calculated the ratio (h-h')/h, which was equal to (2M)/(m+M). The value we obtained was:
(2M)/(m+M) = 1.168
To calculate the fractional loss of kinetic energy, we use the formula:
fractional loss of kinetic energy = (initial kinetic energy - final kinetic energy) / initial kinetic energy
We know that the initial kinetic energy of the projectile before the collision is:
KEi = (1/2)mv² where m is the mass of the projectile and v is its initial velocity.
From Data Table 2, we have:
m = 0.023 kg
v = 310.5 m/s
KEi = (1/2)(0.023)(310.5)² = 1104.4 J
After the collision, the projectile and the pendulum move together with a velocity v', which we calculated in Data Table 2.
The final kinetic energy of the combined system is:
KEf = (1/2)(m+M)v'²
Substituting the values from Data Table 2, we get:
KEf = (1/2)(0.395)(4.34)² = 3.06 J
Therefore, the fractional loss of kinetic energy is:
fractional loss of kinetic energy = (1104.4 - 3.06)/1104.4 = 0.997
We can express the fractional loss of kinetic energy in symbol form as:
fractional loss of kinetic energy = (KEi - KEf)/KEi
Now, let's use the equations from the lab to show that this quantity is equal to M/(m+M).
From the conservation of momentum, we know that:
mvi = (m+M)v'
where vi is the initial velocity of the projectile before the collision.
Solving for v', we get:
v' = (mvi)/(m+M)
The total energy of the combined system before the collision is:
Ei = (1/2)(m+M)v'²
Substituting the expression for v', we get:
Ei = (1/2)(m+M)(mvi)²/(m+M)² = (1/2)mv²/(1+m/M)
where we have used the fact that mvi = mv.
The total energy of the combined system after the collision is:
Ef = (1/2)(m+M)v'² = (1/2)(m+M)(mvi)²/(m+M)² = (1/2)mv²/(1+m/M)
where we have used the expression for v' obtained earlier.
Therefore, the fractional loss of kinetic energy is:
fractional loss of kinetic energy = (Ei - Ef)/Ei = M/(m+M)
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an ideal gas expands from 28.0 l to 92.0 l at a constant pressure of 1.00 atm. then, the gas is cooled at a constant volume of 92.0 l back to its original temperature. it then contracts back to its original volume without changing temperature. find the total heat flow, in joules, for the entire process.
The event of energy being converted into particles and antiparticles occurred when the universe was less than one second old. During this time, the universe was a hot, dense soup of particles, including quarks, leptons, and photons.
The universe began with the Big Bang, which occurred approximately 13.8 billion years ago. At this time, the universe was a hot, dense soup of particles, including quarks, leptons, and photons. The first event to occur after the Big Bang was the conversion of energy into particles and antiparticles. This process, known as particle-antiparticle annihilation, occurred when the universe was less than one second old. Next, protons and neutrons fused to form nuclei such as deuterium and helium. This process, known as nucleosynthesis, occurred when the universe was between one and three minutes old. After nucleosynthesis, the universe consisted of a hot, dense plasma of charged particles. Over time, the universe expanded and cooled, allowing electrons to settle down around nuclei and form neutral atoms. This process, known as recombination, occurred when the universe was approximately 380,000 years old.
Once recombination occurred, the universe became transparent to radiation, allowing light to travel freely through space. This radiation is known as the cosmic microwave background and is observed today as a faint glow in the sky. Finally, stars and galaxies began to form from the clumps of matter that had been created during nucleosynthesis. The first stars are thought to have formed when the universe was approximately 100 million years old. The Milky Way galaxy, which contains our solar system, is estimated to have formed about 13.6 billion years ago, making it one of the oldest galaxies in the universe.
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The total heat flow for the entire process is zero. This is because the process is a closed cycle, where the gas expands and cools, then contracts back to its original volume without any change in temperature.
To explain further, during the first stage of the process where the gas expands from 28.0 l to 92.0 l at a constant pressure of 1.00 atm, the gas does work on its surroundings and absorbs heat from its surroundings to maintain a constant temperature. This is known as an isothermal process.
During the second stage, where the gas is cooled at a constant volume of 92.0 l back to its original temperature, the gas releases heat to its surroundings to maintain a constant volume. This is known as an isochoric process.
During the final stage of the process, where the gas contracts back to its original volume without changing temperature, the gas does work on its surroundings and releases heat to maintain a constant temperature. This is known as an isothermal process.
Since the process is a closed cycle, the total work done by the gas is equal to the total heat absorbed and released by the gas. Therefore, the total heat flow for the entire process is zero.
The total heat flow for the entire process is zero because the process is a closed cycle and the work done by the gas is equal to the heat absorbed and released by the gas.
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a 1.0-ma current of 1.6-mev protons strikes a 2.6-mev-high potential barrier 2.8 x 10-13 m thick. estimate the transmitted current.
The estimated transmitted current is 0.10 mA.
What is Proton?A proton is a subatomic particle found in the nucleus of an atom. It has a positive electric charge and its mass is approximately 1 atomic mass unit (amu). Protons are one of the building blocks of matter and determine the atomic number and chemical properties of an element.
The transmission probability of the protons through the barrier can be calculated using the formula:
[tex]T = e^{(-2kd)[/tex]
where T is the transmission probability, k is the wavevector of the protons, and d is the thickness of the barrier.
The wavevector of the protons can be calculated using the de Broglie relation:
λ = h/p
where λ is the de Broglie wavelength, h is the Planck constant, and p is the momentum of the protons.
Substituting the values given in the problem, we get:
λ = h/p = h/(mv) = (6.626 x 10⁻³⁴ J.s)/[(1.67 x 10⁻²⁷ kg)(1.6 x 10⁶ m/s)] ≈ 2.4 x 10⁻¹⁵ m
The wavevector is then:
k = 2π/λ = 2π/(2.4 x 10⁻¹⁵ m) ≈ 2.6 x 10¹⁵ m⁻¹
Substituting the values of k and d into the formula for transmission probability, we get:
[tex]T = e^{(-2kd)} = e^{[-2(2.6 x 10^{15} m^{-1})(2.8 x 10^{-13 m)]}[/tex] ≈ 0.10
Therefore, the transmitted current is:
[tex]I_{transmitted[/tex] = T x [tex]I_{incident[/tex] = (0.10)(1.0 mA) = 0.10 mA
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A current-carrying rectangular coil of wire is placed in a magnetic field. The magnitude of the torque on the coil is NOT dependent upon which one of the following quantities?
(a) the direction of the current in the loop
(b) the magnitude of the current in the loop
(c) the area of the loop
(d) the orientation of the loop
(e) the magnitude of the magnetic field
The magnitude of the torque on the coil is NOT dependent upon (b) the magnitude of the current in the loop.
Understanding the torque on the coilThe torque on the coil is directly proportional to the product of the magnetic field strength and the area of the loop, as well as the sine of the angle between the magnetic field and the normal to the loop.
The direction of the current in the loop, the area of the loop, the orientation of the loop, and the magnitude of the magnetic field all affect the angle between the magnetic field and the normal to the loop, but not the magnitude of the torque.
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In an integrated circuit, the current density in a2.2-μm-thick × 80-μm-wide gold film is 7.9×105 A/m2 .How much charge flows through the film in 15 min?
The 125.1 coulombs of charge flow through the gold film in 15 minutes.
We can use the equation for current density (J) to find the current (I) flowing through the gold film:
J = I/A
where A is the cross-sectional area of the gold film, given by:
A = t x w
Substituting the given values, we get:
[tex]A = (2.2 \times 10^{-6} m) \times (80 \times 10^{-6} m) = 1.76 \times 10^{-7} m^2I = J \times A = (7.9 \times 10^5 A/m^2) \times (1.76 \times 10^{-7} m^2) = 0.139 AQ = I \times t[/tex]
Substituting the given values, we get:
Q = (0.139 A) x (15 x 60 s) = 125.1 C
Coulombs is named after the French physicist Charles-Augustin de Coulomb who discovered Coulomb's law, which describes the electrostatic interaction between electrically charged particles. Coulombs are used to measure the amount of electric charge in a system, such as in a capacitor or in an electric current.
The Coulomb is an essential unit of measurement in fields such as electrical engineering, physics, and electronics. It is used to quantify the amount of charge that is involved in a wide range of electrical phenomena, including the attraction or repulsion of charged particles, the flow of electricity through a conductor, and the charging of a battery or capacitor.
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How many photons per second enter one eye if you look directly at a 100 W light bulb 2.00 m away? Assume a pupil diameter of 4.00 mm and a wavelength of 600 nm. How many photons per second enter your eye if a 1.00 m W laser beam is directed into your eye? λ=633nm)
The number of photons per second that enter the eye can be calculated using the formula:
N = (P / A) x (t / h) x (1 / E)
where:
P = power of the light source (in watts)
A = area of the pupil (in square meters)
t = transmission coefficient of the cornea and lens (assumed to be 0.95)
h = Planck's constant (6.626 x 10[tex]^-34[/tex] joule-seconds)
E = energy per photon (in joules)
For the 100 W light bulb:
P = 100 W
A = π (0.002 m)^2 = 1.2566 x 10[tex]^-5 m^2[/tex] (assuming the pupil is circular)
t = 0.95 (given)
h = 6.626 x 10[tex]^-34[/tex] J·s (given)
λ = 600 nm = 6.00 x 10[tex]^-7 m[/tex] (given)
c = speed of light = 3.00 x 10m/s (assumed)
E = hc / λ = (6.626 x 10[tex]^-34[/tex] J·s) x (3.0[tex]^8[/tex]0 x 10[tex]^8[/tex] m/s) / (6.00 x 10[tex]^-7 m[/tex]) = 3.31 x 10[tex]^-19[/tex] J
Plugging in the values:
N = (100 W / 1.2566 x 10[tex]^-5 m^2[/tex]) x (0.95) x (1 s / 6.626 x 10[tex]^-34[/tex] J·s) x (1 / 3.31 x 10[tex]^-19[/tex] J)
= 7.70 x 10^16 photons/s
Therefore, about 7.70 x 10[tex]^16[/tex] photons per second enter one eye when looking directly at a 100 W light bulb from a distance of 2.00 m.
For the 1.00 mW laser beam:
P = 1.00 x 10[tex]^-3[/tex] W
A = π (0.002 m[tex])^2[/tex] = 1.2566 x 10[tex]^-5 m^2[/tex] (assuming the pupil is circular)
t = 0.95 (given)
h = 6.626 x 10[tex]^-34[/tex]J·s (given)
λ = 633 nm = 6.33 x 10[tex]^-7[/tex] m (given)
c = speed of light = 3.00 x 10[tex]^8[/tex] m/s (assumed)
E = hc / λ = (6.626 x 10[tex]^-34[/tex] J·s) x (3.00 x 10[tex]^8[/tex]m/s) / (6.33 x 10[tex]^-7[/tex]m) = 3.14 x 10[tex]^-19[/tex] J
Plugging in the values:
N = (1.00 x 10[tex]^-3[/tex]W / 1.2566 x 10[tex]^-5 m^2[/tex]) x (0.95) x (1 s / 6.626 x 10[tex]^-34[/tex] J·s) x (1 / 3.14 x 10[tex]^-19[/tex]J)
= 7.17 x 10^[tex]12[/tex] photons/s
Therefore, about 7.17 x 10[tex]^12[/tex] photons per second enter your eye if a 1.00 mW laser beam with a wavelength of 633 nm is directed into your eye.
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The number of photons per second that enter the eye can be calculated using the formula:
N = (P / A) x (t / h) x (1 / E)
where:
P = power of the light source (in watts)
A = area of the pupil (in square meters)
t = transmission coefficient of the cornea and lens (assumed to be 0.95)
h = Planck's constant (6.626 x 10[tex]^-34[/tex] joule-seconds)
E = energy per photon (in joules)
For the 100 W light bulb:
P = 100 W
A = π (0.002 m)^2 = 1.2566 x 10[tex]^-5 m^2[/tex] (assuming the pupil is circular)
t = 0.95 (given)
h = 6.626 x 10[tex]^-34[/tex] J·s (given)
λ = 600 nm = 6.00 x 10[tex]^-7 m[/tex] (given)
c = speed of light = 3.00 x 10m/s (assumed)
E = hc / λ = (6.626 x 10[tex]^-34[/tex] J·s) x (3.0[tex]^8[/tex]0 x 10[tex]^8[/tex] m/s) / (6.00 x 10[tex]^-7 m[/tex]) = 3.31 x 10[tex]^-19[/tex] J
Plugging in the values:
N = (100 W / 1.2566 x 10[tex]^-5 m^2[/tex]) x (0.95) x (1 s / 6.626 x 10[tex]^-34[/tex] J·s) x (1 / 3.31 x 10[tex]^-19[/tex] J)
= 7.70 x 10^16 photons/s
Therefore, about 7.70 x 10[tex]^16[/tex] photons per second enter one eye when looking directly at a 100 W light bulb from a distance of 2.00 m.
For the 1.00 mW laser beam:
P = 1.00 x 10[tex]^-3[/tex] W
A = π (0.002 m[tex])^2[/tex] = 1.2566 x 10[tex]^-5 m^2[/tex] (assuming the pupil is circular)
t = 0.95 (given)
h = 6.626 x 10[tex]^-34[/tex]J·s (given)
λ = 633 nm = 6.33 x 10[tex]^-7[/tex] m (given)
c = speed of light = 3.00 x 10[tex]^8[/tex] m/s (assumed)
E = hc / λ = (6.626 x 10[tex]^-34[/tex] J·s) x (3.00 x 10[tex]^8[/tex]m/s) / (6.33 x 10[tex]^-7[/tex]m) = 3.14 x 10[tex]^-19[/tex] J
Plugging in the values:
N = (1.00 x 10[tex]^-3[/tex]W / 1.2566 x 10[tex]^-5 m^2[/tex]) x (0.95) x (1 s / 6.626 x 10[tex]^-34[/tex] J·s) x (1 / 3.14 x 10[tex]^-19[/tex]J)
= 7.17 x 10^[tex]12[/tex] photons/s
Therefore, about 7.17 x 10[tex]^12[/tex] photons per second enter your eye if a 1.00 mW laser beam with a wavelength of 633 nm is directed into your eye.
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n moles of an ideal diatomic gas with internal energy E = =nRT are taken through the cyclic process shown on the P-V diagram where P1=2P3 and V2=2V3. V2 a) What are the values of work W, change in internal energy AEint, and heat transfer Q in process 1-2? Express your answers in terms of P3 and V3. b) What are the values of work W, change in internal energy AEint, and heat transfer Q in process 2-3? Express your answers in terms of P3 and V3. c) What are the values of work W, change in internal energy AEint, and heat transfer Q in process 3-1? Express your answers in terms of P3 and V3. d) Calculate the efficiency of the cycle.
a. Therefore, we have: ΔEint = 2P3(V1 - 2V3).
b. Q = ΔEint = (4/5)nCv(T2 - T3) = (4/5)nR(T2 - T3).
c. Therefore, we have: ΔT = T1 - T3 = (P1V1 - P3V3)/nR = (2P3V1 - P3V3)/nR = P3(V1 - V3)/nR and Q = ΔEint + W = nCv(T1 - T3) + 2P3(V1 - V3)
d. In step two of the process, the diatomic gas expands isobarically from volume V1 to volume V2, then cools isochronally from V2 to V3.
a. The work done in process 1-2 is given by:
W = P1(V2 - V1)
Since P1 = 2P3 and V2 = 2V3, we have:
W = 2P3(2V3 - V1)
The change in internal energy in process 1-2 is given by:
ΔEint = Q - W
Q = P1(V2 - V1) = 2P3(2V3 - V1)
b) In process 2-3, the gas is undergoing an isochoric heating from volume V3 to volume V2, followed by an isobaric compression from volume V2 to volume V1.
The work done in process 2-3 is zero since the volume is constant.
The change in internal energy in process 2-3 is given by:
ΔEint = Q - W
Since the process is isochoric, the heat transfer Q is given by:
Q = ΔEint = nCvΔT = nCv(T2 - T3)
PV = nRT
For a diatomic gas, we have:
Cv = (5/2)R/2 = (5/4)R
Substituting for P and V, we have:
Cv(T2 - T3) = (5/4)nR(T2 - T3) = (5/4)ΔEint
Therefore, we have:
Q = ΔEint = (4/5)nCv(T2 - T3) = (4/5)nR(T2 - T3)
c) In process 3-1, the gas is undergoing an isobaric compression from volume V3 to volume V1, followed by an isochoric heating from volume V1 to volume V2.
The work done in process 3-1 is given by:
W = P1(V1 - V3) = 2P3(V1 - V3)
The change in internal energy in process 3-1 is given by:
ΔEint = Q - W
Process is isochoric, the heat transfer Q is given by:
Q = ΔEint = nCvΔT = nCv(T1 - T3)
ΔT = T1 - T3
From the ideal gas law, we have:
PV = nRT
Substituting for P and V, we have:
T = PV/nR
Therefore, we have:
ΔT = T1 - T3 = (P1V1 - P3V3)/nR = (2P3V1 - P3V3)/nR = P3(V1 - V3)/nR
Q = ΔEint + W = nCv(T1 - T3) + 2P3(V1 - V3)
d) The efficiency of the cycle is given by:
η = (Wnet / QH) x 100%
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the magnetic field 41.0 cm away from a long, straight wire carrying current 6.00 a is 2930 nt. (a) at what distance is it 293 nt?
At a distance of 410 cm from the wire, the magnetic field is 293 nT, Straight wire carrying current 6.00 a is 2930 nt.
To answer this question, we will use the formula for the magnetic field B around a straight wire carrying current I:
B = (μ₀ * I) / (2 * π * d)
where μ₀ is the permeability of free space (4π × 10⁻⁷ T·m/A), I is the current, and d is the distance from the wire.
Given the initial magnetic field B₁ = 2930 nT, the current I = 6.00 A, and distance d₁ = 41.0 cm, we can calculate the distance d₂ where the magnetic field is B₂ = 293 nT.
We first find the ratio of the magnetic fields:
B₁ / B₂ = 2930 nT / 293 nT = 10
Since the magnetic field is inversely proportional to the distance, the ratio of distances is:
d₂ / d₁ = 10
Now, we can solve for d₂:
d₂ = 10 * d₁ = 10 * 41.0 cm = 410 cm
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as an admirer of thomas young, you perform a double-slit experiment in his honor. you set your slits 1.03 mm apart and position your screen 3.93 m from the slits. although young had to struggle to achieve a monochromatic light beam of sufficient intensity, you simply turn on a laser with a wavelength of 631 nm . how far on the screen are the first bright fringe and the second dark fringe from the central bright fringe? express your answers in millimeters.
The second dark fringe is approximately 4.85 mm from the central bright fringe. We can use the formula d(sinθ) = mλ to calculate the position of the bright and dark fringes. First, we need to calculate the distance between the slits and the screen in meters:
3.93 m
Next, we need to calculate the distance between the slits:
1.03 mm = 0.00103 m
We can use this distance as the distance between the two sources (the two slits).
The wavelength of the laser is given as:
631 nm = 0.000631 m
We will use this value for λ.
Now we can calculate the angle θ for the first bright fringe:
m = 1 (since we're looking for the first bright fringe)
d = 0.00103 m
λ = 0.000631 m
θ = sin⁻¹(mλ/d)
θ = sin⁻¹(0.000631/0.00103)
θ ≈ 0.617 radians
To find the position of the first bright fringe on the screen, we multiply θ by the distance between the slits and the screen:
x = θd
x = 0.617 x 3.93
x ≈ 2.43 mm
So the first bright fringe is approximately 2.43 mm from the central bright fringe.
To find the position of the second dark fringe, we use the same formula but with m = 2:
θ = sin⁻¹(2λ/d)
θ ≈ 1.235 radians
x = θd
x = 1.235 x 3.93
x ≈ 4.85 mm
So the second dark fringe is approximately 4.85 mm from the central bright fringe.
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a wire 2.22 m long carries a current of 10.7 a and makes an angle of 40.4° with a uniform magnetic field of magnitude b = 1.99 t. calculate the magnetic force on the wire.
Answer:
47.86 N
Explanation:
The magnetic force on a current-carrying wire in a magnetic field is given by the formula:
F = BIL sinθ
where F is the magnetic force, B is the magnetic field, I is the current, L is the length of the wire, and θ is the angle between the wire and the magnetic field.
Substituting the given values, we get:
F = (1.99 T) x (10.7 A) x (2.22 m) x sin(40.4°)
F = 47.86 N
Therefore, the magnetic force on the wire is 47.86 N, in the direction perpendicular to both the magnetic field and the wire.
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A sump pump is draining a flooded basement at the rate of 0.600 L/s, with an output pressure of 3.00 ? 105 N/m2. Neglect frictional losses in both parts of this problem.
(a) The water enters a hose with a 3.00 cm inside diameter and rises 2.50 m above the pump. What is its pressure at this point?
_____N/m2
(b) The hose then loses 1.80 m in height from this point as it goes over the foundation wall, and widens to 4.00 cm diameter. What is the pressure now?
_____N/m2
a. The pressure of the water that enters a hose with a 3.00 cm inside diameter and rises 2.50 m above the pump is 276,475 N/m².
b. The pressure after losing 1.80 m in height and widening to 4.00 cm diameter is 293,715 N/m².
To find the pressure of the water 2.50 m above the pump, we need to account for the change in potential energy. The pressure at this point can be calculated using the following formula:
P2 = P1 - ρgh
where P1 is the initial pressure (3.00 × 10^5 N/m²), ρ is the density of water (approximately 1000 kg/m³), g is the acceleration due to gravity (9.81 m/s²), and h is the height difference (2.50 m).
P2 = 3.00 × 10⁵ N/m₂ - (1000 kg/m³)(9.81 m/s²)(2.50 m)
P2 ≈ 276,475 N/m²
The pressure of the water 2.50 m above the pump is approximately 276,475 N/m².
To find the pressure after losing 1.80 m in height and widening to 4.00 cm diameter, we can use the same formula, adjusting the height difference accordingly:
P3 = P2 + ρgh'
where h' is the new height difference (1.80 m).
P3 = 276,475 N/m² + (1000 kg/m³)(9.81 m/s²)(1.80 m)
P3 ≈ 293,715 N/m²
The pressure after losing 1.80 m in height and widening to 4.00 cm diameter is approximately 293,715 N/m².
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Consider an asteroid with a radius of 17 km and a mass of 3.1×1015 kg . Assume the asteroid is roughly spherical. Suppose the asteroid spins about an axis through its center, like the Earth, with a rotational period T. What is the smallest value T can have before loose rocks on the asteroid's equator begin to fly off the surface?
ga is 7.4*10^-4
The smallest value of T for which loose rocks on the equator of the asteroid begin to fly off its surface is about 13.6 hours.
How can we determine the minimum value of T for which loose rocks on the equator of the asteroid begin to fly off its surface?To solve this problem, we need to find the centrifugal force acting on a rock located on the equator of the asteroid due to its rotation. If the centrifugal force is greater than the gravitational force holding the rock on the asteroid's surface, the rock will fly off the surface.
The centrifugal force is given by:
F = mω²r
where m is the mass of the rock, ω is the angular velocity (i.e., 2π/T), and r is the distance from the rock to the axis of rotation. We want to find the minimum value of T for which the centrifugal force exceeds the gravitational force.
The gravitational force holding the rock on the surface is given by:
Fg = GmM/R²
where G is the gravitational constant, M is the mass of the asteroid, and R is its radius. We can assume that R is much larger than the radius of the rock, so we can use R as the distance from the rock to the center of the asteroid.
Setting F = Fg, we have:
mω²r = GmM/R²
Simplifying, we get:
ω²r = GM/R³
Solving for T, we get:
T = 2π√(R³/GM)
Substituting the given values, we get:
T = 2π√((17 km)³/(6.67×10^-11 Nm²/kg² × 3.1×10^15 kg))
T ≈ 13.6 hours
Therefore, the smallest value of T for which loose rocks on the equator of the asteroid begin to fly off its surface is about 13.6 hours.
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The smallest value of T for which loose rocks on the equator of the asteroid begin to fly off its surface is about 13.6 hours.
How can we determine the minimum value of T for which loose rocks on the equator of the asteroid begin to fly off its surface?To solve this problem, we need to find the centrifugal force acting on a rock located on the equator of the asteroid due to its rotation. If the centrifugal force is greater than the gravitational force holding the rock on the asteroid's surface, the rock will fly off the surface.
The centrifugal force is given by:
F = mω²r
where m is the mass of the rock, ω is the angular velocity (i.e., 2π/T), and r is the distance from the rock to the axis of rotation. We want to find the minimum value of T for which the centrifugal force exceeds the gravitational force.
The gravitational force holding the rock on the surface is given by:
Fg = GmM/R²
where G is the gravitational constant, M is the mass of the asteroid, and R is its radius. We can assume that R is much larger than the radius of the rock, so we can use R as the distance from the rock to the center of the asteroid.
Setting F = Fg, we have:
mω²r = GmM/R²
Simplifying, we get:
ω²r = GM/R³
Solving for T, we get:
T = 2π√(R³/GM)
Substituting the given values, we get:
T = 2π√((17 km)³/(6.67×10^-11 Nm²/kg² × 3.1×10^15 kg))
T ≈ 13.6 hours
Therefore, the smallest value of T for which loose rocks on the equator of the asteroid begin to fly off its surface is about 13.6 hours.
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Consider an absorbing. continuous-time Markov chain with possibly more than one absorbing states. (a) Argue that the continuous-time chain is absorbed in state a if and only if the embedded discrete-time chain is absorbed in state a. (b) Let 1 2 3 4 5 1(0 0 0 0 0 2 1 -3 2 0 0 2-3 0 2 4 20 4 0 0 2 -5 3 50 0 0 0 0 be the generator matrix for a continuous-time Markov chain. For the chain started in state 2, find the probability that the chain is absorbed in state 5
A). The continuous-time chain is absorbed in state an if and only if the embedded discrete-time chain is absorbed in state a.
(b) The probability that the chain is absorbed in state 5, given that it started in state 2, is 20/3.
[tex]N = (I-Q)^{-1},[/tex]
Q = 1 -3 2 0 0
2 -3 0 2 0
0 0 0 0 0
0 0 0 0 0
R = 2 0 0
0 4 2
0 0 50
I = 1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
Therefore, the fundamental matrix N is given by:
[tex]N = (I-Q)^{-1},[/tex]= 1.25 0.75 -0.5 0 0
2.5 3.5 -1.5 -2 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
A continuous-time chain is a mathematical model used to describe the behavior of a system that changes over time. It is a stochastic process that consists of a sequence of random variables, where each variable represents the state of the system at a specific time. The chain evolves in continuous time, meaning that the state of the system can change at any point in time, not just at discrete time intervals.
Continuous-time chains are used in many fields, including physics, biology, finance, and engineering, to model a wide range of phenomena such as the movement of particles in a fluid, the spread of disease in a population, or the behavior of financial markets. The behavior of a continuous-time chain can be analyzed using techniques from probability theory and stochastic processes, such as Markov chains, differential equations, and stochastic calculus.
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Suppose a 25 mH inductor has a reactance of 95 S2. What would the frequency be in Hz? Grade Summary 0% 100% sin0 cosO cotanO asin acosO atanOacotan) sinh0 cosh0tanhO cotanh0 Degrees Radians
The frequency in Hz, given the reactance and the inductor value would be approximately 605.11 Hz.
Reactance (X_L) = 2 * pi * frequency (f) * inductance (L)
In this case, the reactance (X_L) is 95 Ω and the inductance (L) is 25 mH (0.025 H). We can rearrange the formula to solve for the frequency (f):
Frequency (f) = Reactance (X_L) / (2 * pi * inductance (L))
Now, plug in the given values:
Frequency (f) = 95 Ω / (2 * pi * 0.025 H)
Calculate the result:
Frequency (f) ≈ 605.11 Hz
So, the frequency would be approximately 605.11 Hz.
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two long, parallel wires are separated by 4.45 cm and carry currents of 1.73 a and 3.57 a , respectively. find the magnitude of the magnetic force that acts on a 2.13 m length of either wire.
The magnitude of the magnetic force that acts on a 2.13 m length of two long, parallel wires are separated by 4.45 cm and carry currents of 1.73 A and 3.57 A, respectively is 3.64 × 10⁻⁴ N.
To calculate the magnetic force acting on either wire, we can use the formula:
F = (μ₀ × I₁ × I₂ × L) / (2 × π × d)
Where F is the magnetic force, μ₀ is the permeability of free space (4π × 10⁻⁷ T·m/A), I₁ and I₂ are the currents in the wires, L is the length of the wire, and d is the distance between the wires.
Plugging in the given values, we have:
F = (4π × 10⁻⁷ T·m/A × 1.73 A × 3.57 A × 2.13 m) / (2 × π × 0.0445 m)
Calculating the magnetic force, we get:
F ≈ 3.64 × 10⁻⁴ N
So, the magnitude of the magnetic force that acts on a 2.13 m length of either wire is approximately 3.64 × 10⁻⁴ N.
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An elevator weighing 2400 N ascends at a constant speed of 7.0 m/s. How much power must the motor supply to do this?
The motor must supply 16.8kW of power.
Explain power.
The quantity of energy transferred or transformed per unit of time is known as power. The watt, or one joule per second, is the unit of power in the International System of Units. Power is also referred to as activity in ancient writings. A scalar quantity is power.
Power is the pace at which work is completed or energy is delivered; it can be expressed as the product of work completed (W) or energy transferred (E) divided by time (t).
F is 2400N
v is 7m/s
Power will be 2400*7 i.e. 16,800W.
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For a velocity field given by the equation, V = x2yi - y2xj + xyk, determine whether or not this flow field is incompressible. Determine an expression for the vorticity of the flow field described by: V = -xy3i + y4j Is the flow irrotational or rotational? Explain.
The flow is rotational because the curl of the velocity field is nonzero, which implies that there is rotation in the flow. The fact that the vorticity is not zero confirms this.
The divergence of a Gradient vector field V is given by: div(V) = ∂Vx/∂x + ∂Vy/∂y + ∂Vz/∂z.
In this case, the velocity field is given by V = x² y i - y² x j + xy k.
Calculating the divergence:
div(V) = ∂(x² y)/∂x + ∂(-y² x)/∂y + ∂(xy)/∂z
= 2xy - 2yx + 0
= 0
curl(V) = (∂Vz/∂y - ∂Vy/∂z) i + (∂Vx/∂z - ∂Vz/∂x) j + (∂Vy/∂x - ∂x/∂y) k
In this case, Vx = -xy³, Vy = [tex]y^4[/tex], and Vz = 0, so:
curl(V) = (-3y² i - x j) + 0k
The vorticity is the magnitude of the curl, so:
|curl(V)| = √((-3y²)² + x²)
A gradient refers to the rate of change in a variable, typically represented as a slope or derivative. In mathematics, a gradient is a vector that indicates both the direction and magnitude of the greatest rate of change in a function. It is calculated by taking the partial derivatives of the function with respect to each variable and then combining them into a vector.
Gradients are used in a wide range of applications, including optimization problems, computer graphics, and machine learning. In optimization, the gradient is used to find the minimum or maximum value of a function by iteratively adjusting the input variables in the direction of steepest descent or ascent. In computer graphics, gradients are used to create smooth transitions between colors or shades of an image.
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a diffraction grating has 2,160 lines per centimeter. at what angle in degrees will the first-order maximum be for 540 nm wavelength green light? (no response) seenkey 6.7 °
The first-order maximum for 540 nm wavelength green light with a diffraction grating of 2,160 lines per centimeter will be at an angle of 6.7°.
To find the angle of the first-order maximum for a 540 nm wavelength green light with a diffraction grating having 2,160 lines per centimeter, we can use the grating equation,
nλ = d sinθ
where n is the order of maximum (n = 1 for first-order maximum), λ is the wavelength of light (540 nm), d is the distance between the lines (inverse of the number of lines per centimeter), and θ is the angle we want to find.
1. Convert lines per centimeter to distance between lines (d):
d = 1 / 2,160 lines/cm = 1 / (2,160 x 10^2 lines/m) = 1 / 2.16 x 10^5 lines/m
d = 4.63 x 10^-6 m
2. Convert the wavelength from nm to m:
λ = 540 nm = 540 x 10^-9 m
3. Use the grating equation to find the angle θ:
1(540 x 10^-9 m) = (4.63 x 10^-6 m) sinθ
sinθ = (540 x 10^-9 m) / (4.63 x 10^-6 m)
4. Calculate sinθ:
sinθ = 0.1166
5. Find the angle θ:
θ = arcsin(0.1166) = 6.7°
With a 2,160-line-per-centimeter diffraction grating, the first-order maximum for green light with a wavelength of 540 nm will be at an angle of 6.7°.
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design a series rlc type bandpass filter with cutoff frequencies of 10 khz and 12 khz. assuming c = 80 pf, find r, l, and q.
To design a series RLC bandpass filter with cutoff frequencies of 10 kHz and 12 kHz, and C = 80 pF, you need to find R, L, and Q. The values for R, L, and Q are approximately 31.83 ohms, 25.13 μH, and 11.90, respectively.
To determine these values, follow these steps:
1. Calculate the center frequency (f0) and bandwidth (BW) using the given cutoff frequencies:
f0 = (10 kHz + 12 kHz) / 2 = 11 kHz
BW = 12 kHz - 10 kHz = 2 kHz
2. Calculate the filter's quality factor (Q):
Q = f0 / BW = 11 kHz / 2 kHz = 5.5
3. Calculate the inductor value (L) using the center frequency and capacitance:
L = 1 / (4 * π² * f0² * C) ≈ 25.13 μH
where C = 80 pF = 80 * 10⁻¹² F
4. Calculate the resistance (R) using the quality factor, inductor, and center frequency:
R = 2 * π * f0 * L / Q ≈ 31.83 ohms
With these values, you can design a series RLC bandpass filter with the desired characteristics.
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find the index of refraction in a medium in which the speed of light is 2.00 108 m/s.
The index of refraction in a medium is 1.50 in which the speed of light is 2.00 [tex]10^8[/tex] m/s.
The index of refraction of a medium is defined as the ratio of the speed of light in a vacuum to the speed of light in that medium. Therefore, if the speed of light in a medium is 2.00 × [tex]10^8[/tex] m/s, we can find the index of refraction by dividing the speed of light in a vacuum (which is approximately 3.00 × [tex]10^8[/tex]m/s) by the speed of light in the medium:
Index of refraction = speed of light in vacuum / speed of light in medium
Index of refraction = 3.00 × [tex]10^8[/tex]m/s / 2.00 × [tex]10^8[/tex]m/s
Index of refraction = 1.50
Therefore, The index of refraction in a medium is 1.50.
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The index of refraction in a medium is 1.50 in which the speed of light is 2.00 [tex]10^8[/tex] m/s.
The index of refraction of a medium is defined as the ratio of the speed of light in a vacuum to the speed of light in that medium. Therefore, if the speed of light in a medium is 2.00 × [tex]10^8[/tex] m/s, we can find the index of refraction by dividing the speed of light in a vacuum (which is approximately 3.00 × [tex]10^8[/tex]m/s) by the speed of light in the medium:
Index of refraction = speed of light in vacuum / speed of light in medium
Index of refraction = 3.00 × [tex]10^8[/tex]m/s / 2.00 × [tex]10^8[/tex]m/s
Index of refraction = 1.50
Therefore, The index of refraction in a medium is 1.50.
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(a) find the power of the lens necessary to correct an eye with a far point of 26.1 cm
The power of the lens necessary to correct an eye with a far point of 26.1 cm is approximately 3.83 diopters.
To find the power of the lens necessary to correct an eye with a far point of 26.1 cm, we can use the formula:
Power (P) = 1 / focal length (f)
The far point is the distance at which the eye can see clearly. In this case, it is 26.1 cm or 0.261 meters. To correct the vision, the lens should have a focal length equal to the far point.
Focal length (f) = 0.261 meters
Now, we can calculate the power:
P = 1 / 0.261
P ≈ 3.83 diopters
Therefore, a lens with a power of approximately 3.83 diopters is necessary to correct an eye with a far point of 26.1 cm.
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ULF (ultra low frequency) electromagnetic waves, produced in the depths of outer space, have been observed with wavelengths in excess of 29 million kilometers.
Part A
What is the period of such a wave?
According to the question the period of the ULF wave is 10 seconds.
What is period?Period is the term used to describe the monthly cycle of a woman's reproductive system. During each menstrual cycle, a woman's body prepares for pregnancy. The egg is released from the ovary and travels through the Fallopian tubes to the uterus.
We can calculate the period of an ULF wave with the following formula:
Period (T) = 1/Frequency (f)
Since we don't know the exact frequency of the ULF wave, we can calculate an approximate period by using the wavelength (λ) of the wave, which is given as 29 million kilometers. Using the following formula, we can calculate the frequency of the wave:
Frequency (f) = Speed of light (c) / Wavelength (λ)
Substituting the values, we get:
f = 3 x 10⁸ m/s / 29 x 10⁶ km
f = 0.1 Hz
Now, we can calculate the period of the ULF wave using the formula:
T = 1/f
T = 1/0.1
T = 10 s
Therefore, the period of the ULF wave is 10 seconds.
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According to the question the period of the ULF wave is 10 seconds.
What is period?Period is the term used to describe the monthly cycle of a woman's reproductive system. During each menstrual cycle, a woman's body prepares for pregnancy. The egg is released from the ovary and travels through the Fallopian tubes to the uterus.
We can calculate the period of an ULF wave with the following formula:
Period (T) = 1/Frequency (f)
Since we don't know the exact frequency of the ULF wave, we can calculate an approximate period by using the wavelength (λ) of the wave, which is given as 29 million kilometers. Using the following formula, we can calculate the frequency of the wave:
Frequency (f) = Speed of light (c) / Wavelength (λ)
Substituting the values, we get:
f = 3 x 10⁸ m/s / 29 x 10⁶ km
f = 0.1 Hz
Now, we can calculate the period of the ULF wave using the formula:
T = 1/f
T = 1/0.1
T = 10 s
Therefore, the period of the ULF wave is 10 seconds.
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A 26.5 kΩ resistor connected to an AC voltage source dissipates an average power of 0.800 W. HINT (a) Calculate the rms current in the resistor (in A). (b) Calculate the rms voltage of AC source (in V).
(a) The rms current in the resistor is 6.11 mA.
(b) The rms voltage of the AC source is 161.8 V.
To find the rms current (I) in the resistor, we use the formula P = I²R, where P is the average power (0.800 W) and R is the resistance (26.5 kΩ).
Step 1: Rearrange the formula to solve for I: I = √(P/R)
Step 2: Convert the resistance to ohms: 26.5 kΩ = 26500 Ω
Step 3: Plug the values into the formula: I = √(0.800 W / 26500 Ω) = 6.11 x 10⁻³ A, or 6.11 mA.
To find the rms voltage (V) of the AC source, we use the formula V = IR.
Step 4: Plug the values into the formula: V = (6.11 x 10⁻³ A) x (26500 Ω) = 161.8 V.
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What is the average power loss in crab nebula?
The average power loss in the Crab Nebula is estimated to be around 4.6 × 10^38 erg/s, which is equivalent to about 2.2 million times the power output of the sun.
What's Crab NebulaThe Crab Nebula is a supernova remnant that emits radiation across the electromagnetic spectrum. The energy of this radiation is thought to come from the rotational energy of the pulsar at its center.
The power loss is due to the emission of radiation in the form of synchrotron radiation and inverse Compton scattering. These processes are responsible for producing the high-energy gamma-ray emission observed from the Crab Nebula.
Understanding the energy output of the Crab Nebula is important for studying the processes that occur in supernova remnants and for understanding the behavior of pulsars.
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A rod suspended at its end acts as a physical pendulum and swings with a period of 1. 4 s. What is the length of this physical pendulum? Assume that g=9.8 m/s2
The length of this physical pendulum is [tex]1.207\ meters.[/tex] A rod suspended at its end acts as a physical pendulum and swings with a period of [tex]1. 4 s[/tex]
The formula for the period of a physical pendulum is:
[tex]T = 2\pi * \sqrt{L / g}[/tex]
The time period of a pendulum is the time it takes for one complete oscillation or swing. It is the time taken for the pendulum to return to its original starting position after being displaced and released.
where:
T = period of the pendulum,
L = length of the pendulum,
g = acceleration due to gravity,
Now, rearranging the formula to solve for L:
[tex]L = (g / (4\pi^2)) * T^2\\L = (9.8 / (4 * 3.14²)) * (1.4 )^2\\L = (9.8 / 39.478) * 1.96\\L = 1.207\ meters[/tex]
So, the length of this physical pendulum is [tex]1.207\ meters.[/tex]
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A box having a mass of 1.5 kg is accelerated across a table at 1.5 m/s2. The coefficient of friction on the box is 0.3. What is the force being applied to the box? If this force were applied by a spring, what would the spring constant have to be in order for the spring to be stretched to only 0.08 m while pulling the box?
Avechile is being planed that is driven by a fly wheel engine it has to run for at least 30minute and develop teacly power of 500w
How much Energy will fly wheel need to supply?
For high-pass shelving filters: if you run the frequency response high enough, it eventually rolls off. Give two possible sources of a low-pass pole.
For high-pass shelving filter, if you run the frequency response high enough, it eventually rolls off due to the nature of the filter's design. This means that at very high frequencies, the filter will start to attenuate the signal, effectively acting as a low-pass filter.
Two possible sources of a low-pass pole in a high-pass shelving filter include the capacitor and the op-amp. Capacitors have a tendency to act as low-pass filters due to their inherent frequency-dependent impedance. Additionally, op-amps can introduce a low-pass pole into the circuit due to their finite gain bandwidth product and the effect of the feedback network on the circuit's frequency response.
Two possible sources of a low-pass pole are:
1. Parasitic capacitance: Unintended capacitance that forms between components or traces on a circuit board can create a low-pass pole, as it causes the high-frequency signal to be attenuated.
2. Component limitations: The frequency response of active components like op-amps or transistors can limit the bandwidth of a filter. As the frequency increases, these components may not respond quickly enough, resulting in a low-pass pole.
These factors can cause the high-frequency response of a high-pass shelving filter to roll off.
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introduction and conclusion on determining the geometric method of adding vectors using parallelogram method
Introduction:
In physics and mathematics, vectors are quantities that have both magnitude and direction. Adding vectors is an essential operation in vector algebra, and there are different methods to achieve it. One of the most popular ways of adding vectors is the parallelogram method, which involves constructing a parallelogram using the vectors as adjacent sides and then finding the diagonal of the parallelogram.
Body:
The parallelogram method is a geometric method of adding vectors. It works on the principle that if two vectors are represented by adjacent sides of a parallelogram, then their sum is represented by the diagonal of the parallelogram. To use this method, draw two vectors as adjacent sides of a parallelogram, and then draw the diagonal from the initial point of the two vectors to the opposite corner of the parallelogram. The length and direction of the diagonal represent the magnitude and direction of the sum of the two vectors, respectively.
Conclusion:
The parallelogram method is an intuitive and straightforward way of adding vectors. It is particularly useful when dealing with two-dimensional vectors as it requires only basic geometric knowledge. However, it is not the most efficient method, especially when dealing with many vectors in three dimensions. Other methods, such as the component method, may be more appropriate in such cases. Nonetheless, the parallelogram method remains an essential tool in the study of vectors and provides a useful visualization of vector addition.
What are vectors?In mathematics and physics, a vector is a mathematical object that has both magnitude and direction. Geometrically, a vector can be represented as an arrow with a specified length and direction. Vectors are used to represent quantities that have both size and direction, such as velocity, force, and displacement.
They can be added, subtracted, and multiplied by scalar quantities (e.g., numbers) to produce new vectors that represent the resulting magnitude and direction. Vectors play a fundamental role in many areas of mathematics and physics, including calculus, linear algebra, mechanics, and electromagnetism, among others.
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A 70 kg person can achieve the maximum speed of 2.5 m/s while running a 100 m dash. Treat the person as a point particle.
a. At this speed, what is the person's kinetic energy?
Express your answer with the appropriate units.
b. To what height above the ground would the person have to climb in a tree to increase his gravitational potential energy by an amount equal to the kinetic energy you calculated in part A?
Answer:
a. 218.75 J b. 0.3125m
Explanation:
A:
Kinetic energy is found using the formula [tex]KE = \frac{1}{2}*m*v^2[/tex]
Plugging in 2.5 m/s for velocity and 70 kg for mass we get 218.75 J
B:
To find the height the person has to reach for his kinetic energy to be equal to his potential energy you use the equation [tex]PE = m*g*h[/tex] and set the kinetic energy equation equal to the potential energy equations, in which you will get:
[tex]\frac{1}{2}*m*v^2=m*g*h\\ \frac{1}{2}*v^2=g*h\\ h=\frac{v^2}{2g}[/tex]
h = 0.3125 meters
Answer:
a. 218.75 J b. 0.3125m
Explanation:
A:
Kinetic energy is found using the formula [tex]KE = \frac{1}{2}*m*v^2[/tex]
Plugging in 2.5 m/s for velocity and 70 kg for mass we get 218.75 J
B:
To find the height the person has to reach for his kinetic energy to be equal to his potential energy you use the equation [tex]PE = m*g*h[/tex] and set the kinetic energy equation equal to the potential energy equations, in which you will get:
[tex]\frac{1}{2}*m*v^2=m*g*h\\ \frac{1}{2}*v^2=g*h\\ h=\frac{v^2}{2g}[/tex]
h = 0.3125 meters
A 10 kg sack slides down a smooth surface. If the normal force on the surface at the flat spot, A is 98.1 N (↑↑), the radius of the curvature is _____.a. 0.2 mb. 0.4 mc. 1.0 md. None of the above.
The radius of the curvature is 1.0 m (option c) If the normal force on the surface at the flat spot, A is 98.1 N (↑↑) .
To calculate the radius of curvature using this formula:
radius of curvature (r) = (mass × acceleration due to gravity) / normal force
Step 1: Identify the mass (m), acceleration due to gravity (g), and normal force (N).
mass (m) = 10 kg
acceleration due to gravity (g) = 9.81 m/s²
normal force (N) = 98.1 N
Step 2: Plug in the values into the formula.
radius of curvature (r) = (10 kg × 9.81 m/s²) / 98.1 N
Step 3: Perform the calculations.
radius of curvature (r) = (98.1 kg m/s²) / 98.1 N
Step 4: Simplify the result.
radius of curvature (r) = 1 m
So, the radius of the curvature is 1.0 m .Hence, option c is correct.
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