6. An ice skater is spinning at 60 RPM when she pulls in her arms and undergoes an angular acceleration of 10 rad/s"" for 0. 6 seconds. What was her final RPM and how many rotations did she spin during this time?

(a) Equation for the radius of the ball, r, treating it as a sphere, in terms of its surface area will be  [tex]r & =\sqrt{\frac{A}{4 \pi}}[/tex]

(b) The density of the ball is 4.9307 g/cm^3.

(c) The density of the ball is 4903.7 Kg/m^3 or 4904 Kg/m^3

Surface area of a spherical ball, A=45 cm^2

Part a

The surface area of a spherical ball is given as

[tex]$$\begin{aligned}A & =4 \pi r^2 \\\therefore r & =\sqrt{\frac{A}{4 \pi}}\end{aligned}$$[/tex]

So, this is the expression for the radius of the ball in terms of surface area.

Part b

Mass of the ball, M=140 grams

The density of the ball is given as [tex]$$\rho=\frac{M}{V}$$[/tex]

Where, V is the volume of the spherical ball.

And [tex]$V=\frac{4}{3} \pi r^3$[/tex]

So, the value of the density will be

[tex]$$\begin{gathered}\rho=\frac{M}{\frac{4}{3} \pi r^3} \\\rho=\frac{3 M}{4 \pi r^3}\end{gathered}$$[/tex]

Now, putting the value of "r" in the above expression,

We will get

[tex]$$\begin{aligned}& \rho=\frac{3 M}{4 \pi\left(\sqrt{\frac{A}{4 \pi}}\right)^3} \\& \therefore \rho=\frac{3 M}{4 \pi\left(\frac{A}{4 \pi}\right)^{\frac{3}{2}}}\end{aligned}$$[/tex]

Now putting the value of M and A in above,

We will get,

[tex]$$\begin{gathered}\therefore \rho=\frac{3 \times 140}{4 \pi\left(\frac{45}{4 \pi}\right)^{\frac{3}{2}}} \\\therefore \rho=\frac{420}{4 \times 3.14(3.583)^{\frac{3}{2}}} \\\therefore \rho=\frac{420}{4 \times 3.14 \times 6.782} \\\therefore \rho=\frac{420}{85.18} \mathrm{~g} / \mathrm{cm}^3 \\\therefore \rho=4.9307 \mathrm{~g} / \mathrm{cm}^3\end{gathered}$$[/tex]

Part c

The density in Kg/m^3 is to be calculated.

[tex]$1 \mathrm{~g} / \mathrm{cm}^3=1000 \mathrm{~kg} / \mathrm{m}^3$[/tex]

So,

[tex]$$\begin{gathered}\therefore 4.9307 \mathrm{~g} / \mathrm{cm}^3=1000 \times 4.9307 \mathrm{~kg} / \mathrm{m}^3 \\\therefore \rho=4903.7 \mathrm{~kg} / \mathrm{m}^3\end{gathered}$$[/tex]

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Work done on a 1500kg elevator by its cable to lift it 40m at a constant speed is 592000N.

The work done by a force is calculated as the product of the force(F) and the displacement of the object on which the force is acted.

Work done = F×d

downward forces are weight of lift and frictional force

upward force is only tension of string.

T = Frictional force + Mg

Mg = 1500× 9.8 = 14700N

given frictional force is 100N

So, Tension in string = (100+ 14700) N

= 14800N

work done by tension in lifting = tension force ×

displacement of lift

= 14800 × 40

= 592000N.

hence, work done on a 1500 kg elevator by its cable to lift it 40m at a constant speed is 592000N.

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The terrestrial planets have small size, high density, and are closest to the Sun because the extremely high temperatures associated with formation allowed only silicate minerals and metals to form solids.

The substance's mass per unit of volume is its density (also known as specific mass or volumetric mass density). The sym Density and mass concentration have the same numerical value for a pure substance. Density can be important to packaging, purity, and buoyancy because different materials typically have varied densities. At conventional settings for temperature and pressure, osmium and iridium are the densest elements currently known. The Greek letter rho, written in lower case, is the bol most frequently used to denote density, while the Latin letter D is also acceptable. According to mathematics, density is determined by dividing the mass by the volume: where is the density, m is the mass, and V is the volume. In some circumstances, such as in the oil and gas business in the United States, density is merely defined as something's weight.

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the intensity level of the combined sounds is 67.76 dB

Calculation :

Sound Intensity level L= 63 dB

We know L = 10 log(I/Io)

63 = 10 log(I/Io)

6.3 = log (I/Io)

I /Io = [tex]10^{6.3}[/tex]

I = [tex]10^{6.3}[/tex] Io

= [tex]10 ^{6.3 }[/tex]x [tex]10 ^{-12}[/tex]

= [tex]10^{ -5.7}[/tex] W/m 2

  = 1.995 x[tex]10 ^{-6}[/tex] W/m 2

Sound Intensity level L '= 66 dB

We know L ' = 10 log(I '/Io)

66 = 10 log(I '/Io)

6.6 = log (I'/Io)

I ' /Io = [tex]10 ^{6.6}[/tex]

I ' =  [tex]10 ^{6.6}[/tex] Io

=  [tex]10 ^{6.6}[/tex] x [tex]10 ^{-12}[/tex]

= [tex]10 ^{-5.4}[/tex]W/m ^2

  = 3.981 x[tex]10 ^{-6}[/tex] W/m^ 2

Combined intensity I " = I + I ' = 5.976 x10 -6 W/m 2

the intensity level of the combined sounds L " = 10 log(I" / Io)

 L " = 10 log [( 5.976 x[tex]10 ^{-6}[/tex] )/([tex]10 ^{-12}[/tex] ) ]

      = 10 log [5.976 x10 6]

      = 10(6.776)

      = 67.76 dB

Intensity is the amount of energy a wave transports in one unit of area per unit time and is also equal to the energy density multiplied by the wave velocity. Usually measured in watts per square meter. Intensity depends on wave strength and amplitude.

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The initial speed of the car with a length of the skid mark of about 64 m long and kinetic friction of 0.70 is 29.63 m/s.

A force that acts between moving surfaces is referred to as kinetic or sliding friction. It is the frictional force that prevents relative motion between surfaces in contact. The formula to calculate this kind of friction is, [tex]f _k =\mu_kN[/tex] where uk is the coefficient of friction and N is the normal force.

Given the length of the skid mark (Δd) is 64 m and the coefficient of kinetic friction is 0.70. The final velocity vf is 0 because the car stops.

Let us assume the initial velocity vi, mass m,  and deceleration a. Then, the normal force acting on the car is N = mg. The kinetic friction will be,

[tex]f _k =\mu_kN\\f_k=\mu_kmg[/tex]

Let's assume the only force acting on the system is kinetic friction, then friction force=ma. Solve this equation with the above equation,

[tex]\begin{aligned}ma&=\mu_kmg\\a&=\mu_kg\end{aligned}[/tex]

Then, the initial speed is calculated as follows

[tex]\begin{aligned}v_f^2&=v_i^2-2a\Delta d\\0&=v_i^2-2\mu_kg\Delta d\\v_i^2&=\mathrm{2\times0.70\times9.80\;m/s^2\times64\;m}\\v_i^2&=\mathrm{878.08\;m^2/s^2}\\v_i&=\mathrm{29.63\;m/s}\end{aligned}[/tex]

The answer is 29.63 m/s.

The complete question is -

Police investigators, examining the scene of an accident involving two cars, measured 64 m long skid marks on one of the cars, which nearly came to a stop before colliding. The coefficient of kinetic friction between rubber and the pavement is about 0.70. Estimate the initial speed of that car, assuming a level road.

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The pear will not hit the fish where he eyes the fish in the water.

Please note that fish in the water will not be the same as what our eyes see. Where the location of the position will be different because it follows the refraction of light obtained so that it can be concluded that the position of the fish is not the actual position. The fish swims forward and the shooter looks at the water from above as if the fish is looking up.

This event is caused by the process of refraction of light. Refraction is the deflection of the direction of propagation of light when it enters a medium of different densities.

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91.66 A is the amount of current flowing through a light bulb with a resistance of 1.2 when it is connected to a 110 volt power source.

current (I) = V/R

= 110 volt / 1.2 ohm

= 91.666666666667 ampere (A)

ions (atoms that have acquired or lost one or more electrons), holes, or subatomic charged particles such as electrons with a negative charge or protons with a positive charge make up electric current (electron deficiencies that may be thought of as positive particles).

Electrons serve as the charge carriers in a wire, and the amount of charge moving through any point of the wire in a given amount of time is measured as electric current. In direct current, the direction of the electric charges is not regularly reversed, although it is in alternating current. In many situations, the flow of positive charge is assumed to be in the same direction as the current in electric circuits.

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In an experiment with n = 14 and preal = 0.10, then the power using a = 0.012 tail will be 0.997

Power is the capacity to shape or direct events, other people's behavior, or both. It is the ability to decide what to do and then do it. It might be either physical—like power—or mental—like wisdom. Power may also be social, political, or economic. It can be employed constructively, such as to effect change and defend human rights, or destructively, such as to repress and exert control. Power may be utilized to both bring about and stop change. Politics, economics, and culture all exhibit it since it is an essential component of all social relations.

N = 14

Mean, mu=0.10,

s=sqrt [0.10(1-0.10)/14]=0.08,

cumulative probability, 0.003,

Power=1-0.003=0.997

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To establish a geosynchronous orbit at a distance of 8.6 105 m, the spacecraft must orbit the planet in the same period as the planet's rotation.

This means that the spacecraft must complete one orbit every T = 31 hours. The orbital period is related to the orbital radius by the equation T2 = 4π2R3/GM, where G is the gravitational constant and M is the mass of the planet.

Substituting in the values for T and R into this equation, we can solve for GM: GM = 4π2(8.6 105 m)3/T2 = 2.08 1017 m3/s2.

The spacecraft must then use its thrusters to adjust its speed and position until it enters an orbit with a period of 31 hours and a radius of 8.6 105 m. It can achieve this by using a combination of delta-v maneuvers to adjust its speed and trajectory. The first maneuver will be a prograde burn to increase the spacecraft's speed until it enters an orbit that is slightly more distant than the desired orbit. The spacecraft will then use a series of impulsive maneuvers to adjust its trajectory until it is in an orbit with the desired radius. Finally, the spacecraft will use a retrograde burn to reduce its speed until it reaches the desired orbital period.

Once the spacecraft has completed all of these maneuvers, it will be in a geosynchronous orbit at a distance of 8.6 105 m from the planet's surface.

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A) f= 2/7mgsin∅ frictional force is exerted on the ball, B) u= 2/7gtan∅ coefficient of friction is required to prevent slipping. A solid ball of inertia m rolls without slipping down a ramp that makes an angle θ with the horizontal.

Part a)

Force equation  is given as

mg sin∅-f= ma

now for torque equation of the ball

f R= 2/5 R²(a/R)

f=2/5 ma

now from above two equations

mg sin∅-2/5 ma=ma

mg sin∅+7/5 ma

a=5/7 g sin∅

so frictional force is given as

f=2/5 ma

f= 2/5m (5/7sin∅)

f=2/5 mg sin∅

Part b)

We also understand that when motion is moving normally, we have

Fn= mg cos∅

so we have

f=uFn

2/7 mg sin∅= u(mg cos∅)

now we have

u= 2/7 g tan∅

A solid ball of inertia m rolls without slipping down a ramp that makes an angle θ with the horizontal. a. What frictional force is exerted on the ball?b. As a function of θ, what coefficient of friction is required to prevent slipping?

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If a = a1, a2, a3 and b = b1, b2, b3 . Then [tex]& \vec{a} \times \vec{b}=(-1)(\vec{b} \times \vec{a})[/tex] Or [tex]& \vec{a} \times \vec{b}=-\vec{b} \times \vec{a}[/tex]

As per the details share in the above question are as bellow,

The provided data are follow,

The product of two or more integers that remain the same, regardless of the sequence in which they are multiplied, is referred to as the commutative property formula for multiplication. The commutative property formula for multiplication is (A x B) = (B x A).

Let us consider, Let a = a1, a2, a3

And b=b1, b2, b3

We have the property that a × b = − b × a.

Then a × b = __ = (−1) __ =− b × a.

[tex]& \vec{a}=a_1 i+a_2 j+a_3 k \\ \vec{b}=b_1 i+b_2 j+b_3 k[/tex]

Then,

[tex]$$\vec{a} \vec{x} \vec{b}=\left|\begin{array}{ccc}i & j & k \\a_1 & a_2 & a_3 \\b_1 & b_2 & b_3\end{array}\right|$$[/tex]

[tex]$$=i\left(a_2 b_3-a_3 b_2\right)+j\left(a_3 b_1-a_1 b_3\right)+k\left(a_1 b_2-a_2 b_1\right)$$[/tex]

So further we get,

[tex]& =i\left(b_2 a_3-a_2 b_3\right)+j\left(a_1 b_3-a_3 b_1\right)+k\left(a_2 b_1-a_1 b_2\right) \\& =-\left[i\left(a_2 b_3-a_3 b_2\right)+j\left(a_3 b_1-a_1 b_3\right)+k\left(a_1 b_2-a_2 b_1\right)\right.[/tex]

So, we get,

[tex]\vec{a} \times \vec{b}=\left[i\left(a_2 b_3-a_3 b_2\right)+j\left(a_3 b_1-a_1 b_3\right)+k\left(a_1 b_2-a_2 b_1\right)\right.[/tex]

We get,

[tex]& \vec{b} \times \vec{a}=-[\vec{a} \times \vec{b}][/tex]

Or [tex]& -\vec{b} \times \vec{a}=\vec{a} \times \vec{b}[/tex]

Then,

[tex]& \vec{a} \times \vec{b}=(-1)(\vec{b} \times \vec{a})[/tex]

Or [tex]& \vec{a} \times \vec{b}=-\vec{b} \times \vec{a}[/tex]

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The temperature of a black body that would radiate most intensely at 560 nanometers (nm) would be approximately 5,379 Kelvin.

Wavelength is the distance between two consecutive peaks (or troughs) of a wave. It is most commonly associated with light and sound waves, but can also refer to electromagnetic radiation, like gamma rays, x-rays, and radio waves. Wavelength is usually measured in meters or nanometers. The shorter the wavelength, the higher the frequency of the wave, and the higher the energy of the wave.

This is because black bodies emit radiation with a peak intensity that is determined by their temperature. The hotter they are, the shorter the wavelength of the peak radiation. The equation used to calculate this is Wien's displacement law, which states that the wavelength of peak radiation is inversely proportional to the temperature of the black body.

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Answer:

The DART mission proved to be successful. Its ultimate goal was to alter the trajectory of the asteroid Dimorphous. It was successful because the asteroid's trajectory was in fact altered. This mission is the first mission that intentionally altered the trajectory of a non man made object in space.

The new (resultant) vector could be written as 0 m, since the vectors cancel each other out. The vector going 5 m West is cancelled out by the vector going 5 m East, and the vector going 11 m West is cancelled out by the vector going 11 m East. This means that the final result is a vector with a magnitude of 0 and no direction.

The accelerating expansion of the universe is thought to be driven by thermal energy.

Acceleration is the charge at which velocity modifications with time, in terms of each pace and path. A factor or an object moving in an instant line is extended if it quickens or slows down. Movement on a circle is expanded even though the speed is steady because the path is always converting.

Mainly, Uniform acceleration, non-uniform acceleration, and average acceleration are the three styles of elevated motions.

Acceleration (a) is the alternate in speed (Δv) over the trade in time (Δt), represented through the equation a = Δv/Δt. This lets you measure how rapidly the pace changes in meters in step with second squared (m/s^2).

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Top-Down: Gas and dark matter collided, causing the gas to star-form. merging of spiral galaxies whose gas has already turned into stars (bottom-up) is the way how the elliptical galaxy formation was happened.

spiral galaxies evolved from bottom-up, and elliptical galaxies from a mix of top-down and bottom-up. There is extremely little raw material for the creation of new stars in elliptical galaxies, in contrast to spiral galaxies, which contain abundant stores of the dust and gas that support star formation. When two spiral galaxies collide, they change from their recognized form into the less organized elliptical galaxies. These old galaxies are considered to be powered by a supermassive black hole. These gas- and dust-eating giants may contribute to the elliptical galaxies' slower development.

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pushed to the right = 300 newtons

pushed to the left = 400 newtons

[tex]400-300=100[/tex]

[tex]\fbox{100 Newtons to the left}[/tex]

The definite integral of this function 60f(x)dx is given by \int{0}^{6} f(x)dx = 18∫06f(x)dx=18.

A definite integral describes the area of a function within the interval formed by the bounds of the definite integral. This means that we can find the definite integral for this problem simply by finding the area of the function.

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The moment of inertia is measured in units of mass times squared distance, such as kgm2. The rotational kinetic energy of an item in rotation about an axis is K = 12I2.

Moment of inertia divided by half equals rotational kinetic energy (angular speed). Where 0 0 is the starting angular velocity, = 0 + t (constant) and = 0 + t (constant). The equation is the same as the linear version, with the exception that the angular equivalents of the linear variables. It is necessary to specify the moment of inertia in relation to a selected axis of rotation. The moment of inertia for a point mass is equal to the mass times the square of the distance perpendicular to the rotation axis, or I = mr2.

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A negative object so it takes in the charge object and nothing will happen until something else would touch it.

Electric charge is the property of a material that makes it feel a force when subjected to an electromagnetic field. Protons and electrons most typically carry positive and negative charges, which are the two forms of electric charges.

The basis of the SI system of physical units is the coulomb It is referred to as C. A coulomb is the unit of power that a one-ampere current can carry in one second.

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Each photon's momentum in the light beam is p=h=632. 8106. The first gas laser was the helium-neon laser.

Although the first laser transitions were in the near infrared, the red laser, with a wavelength of 632.8 nm, is now the most widely employed transition. This laser is offered in lengths ranging from around 10 cm to more than 100. For longer wavelength lines, helium-neon lasers with a 1- to 10-m W output at 543 nm (green), 595 nm (yellow), or 633 nm provide an option to krypton ion lasers (red). A helium-neon laser can produce light with a power output of 1 mW at a wavelength of 632.8 nm.

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The kinetic energy of another body having three times the mass of the given object will be 108 J

Kinetic energy-In physics, the kinetic energy of an object is the energy  it has due to  motion. It is defined as the work required to accelerate a body from a given mass from rest to a specified speed.

The kinetic energy of a body is given by [tex]K.E= \frac{1}{2}Mv^{2}[/tex] (first equation)

Given K.E= 36J

Now the mass of second body is m=3M

also the second body is moving with the same speed v

So kinetic energy of second body K.E₂= [tex]\frac{1}{2}mv^{2} =\frac{1}{2}(3M)v^{2}[/tex]=36×3

=108J (from first equation)

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Nuclear fission is used by all nuclear power plants, and uranium atoms are typically used in nuclear power plants.In nuclear fission, an neutron splits an atom of uranium by slamming against it; two neutrons are created in the process.

A U-235 nucleus immediately splits in half after taking in an additional neutron.Fission is the term for this action (see diagram below).A split U-235 nucleus releases two to three neutrons each time it divides.As a result, starting a chain reaction is possible.

Continuous fission was achieved as more neutrons are successively emitted.Fission-generated neutrons have tremendous energy and move very swiftly.The procedure of moderation is used in most reactors to slow down these so-called rapid neutrons because they do not promote fission as effectively as slower-moving ones do.

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Electric flux through the surface is [tex]1.5 Nm^2/C[/tex].

As per the detail share in the above question are as bellow,

Given an area A with a unit normal vector n, the electric flow through that region is given by Φ = E.(An) = |E|Acosθ

Where angle between vectors E and n =θ .

Also Magnitude of E = |E|

When the magnitude of n is |n| = 1

The normal n and the electric field vector E make an angle of,

θ = 90 - 30 = 60 degrees,

Therefore magnitude of,

|E| = 300N/C

And A= 0.1 x 0.1 = 0.01 m.

Thus,

Φ = |E|Acosθ

Substitute the value in above equation we get,

=300 x 0.01 x cos(60)

=3 x 0.5

[tex]= 1.5 Nm^2/C[/tex]

Electric flux=Φ=[tex]1.5 Nm^2/C[/tex]

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Note : The correct question would be as follow,

what is the electric flux through the surface shown in the figure? assume that e=300n/c.

Elliptical galaxies most likely to be found in the central regions

Galaxies with an elliptical form are known as elliptical galaxies. Since an ellipse has an oval shape, these galaxies also have an oval shape. Elliptical galaxies are those with little to no gas and dust. The elliptical galaxies are the most abundant type of galaxies. Elliptical galaxies are most likely to be found in the central regions of galaxy clusters. Galaxy clusters are large groups of galaxies that are bound together by gravity and are some of the largest structures in the universe.

Elliptical galaxies are characterized by a lack of distinct features, such as spiral arms or a visible nucleus. These galaxies are thought to have formed through the merger of smaller galaxies and are generally found in high-density environments such as the centres of galaxy clusters. They are less common on the outskirts of galaxy clusters and are rarely found in small clusters or in between larger clusters.

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The motion of the free electron sea causes the electrical conduction.

Metals that are near to one another have valence electrons that do not only remain on their own atom but also easily migrate throughout the entire metal complex. They seem to be floating through an sea of electrons, like a solitary water molecule floating freely in the sea. For this reason, it is often referred to as the electron sea model.

These atoms become positively charged cations because each metal atom is free to move around. These cations resemble a positively charged island because they are encircled by a sea of negatively charged electrons. Because of these free-moving electrons, metal is a good conductor of electricity.

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The Sun's surface emits as a black body its surface temperature is [tex]$5.78 \times 10^3 \mathrm{~K}$[/tex].

The sun's surface area is,

A 3D object's surface area is the entire area that all of its faces cover. If we need to figure out how much paint is needed to paint a cube, for example, we may use the cube's surface area. It is consistently expressed in square units.

[tex]A=4 \pi r^2[/tex]

The sun's radius, r, is indicated here.

Substitute [tex]$6.96 \times 10^8 \mathrm{~m}$[/tex] for r.

[tex]A & =4 \pi\left(6.96 \times 10^8 \mathrm{~m}\right)^2 \\& =6.09 \times 10^{18} \mathrm{~m}^2[/tex]

The radiated power is, according to Stefan's law

Energy that emanates from an origin and moves through space at the speed of light is referred to as radiation. This radiation has wave-like qualities and is accompanied by an electric field and a magnetic field.

[tex]P & =\sigma A e T^4 \\T^4 & =\frac{P}{\sigma A e} \\T & =\left(\frac{P}{\sigma A e}\right)^{\frac{1}{4}}[/tex]

Substitute [tex]$6.09 \times 10^{18} \mathrm{~m}^2$[/tex] for A,

[tex]5.67 \times 10^{-8} \mathrm{~W} / \mathrm{m}^2. \mathrm{K}^4$[/tex] .  for [tex]$\sigma$[/tex],

and [tex]$3.85 \times 10^{26} \mathrm{~W}$[/tex] for P.

[tex]T & =\left(\frac{3.85 \times 10^{26} \mathrm{~W}}{\left(5.67 \times 10^{-8} \mathrm{~W} / \mathrm{m}^2 \cdot \mathrm{K}^4\right)\left(6.09 \times 10^{18} \mathrm{~m}^2\right)(1)}\right)^{\frac{1}{4}} \\& =\left(1.12 \times 10^{15} \mathrm{~K}^4\right)^{\frac{1}{4}} \\& =5.78 \times 10^3 \mathrm{~K}[/tex]

The temperature is [tex]$5.78 \times 10^3 \mathrm{~K}$[/tex].

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Note:- The correct question would be as follow,

The radius of our Sun is [tex]$6.96 \times 10^8 \mathrm{~m}$[/tex], and its total power output is [tex]$3.85 \times 10^{26} \mathrm{~W}$[/tex]. (a) Assuming the Sun's surface emits as a black body, calculate its surface temperature.

If the coil is then rotated through an angle of 30o in a time of 0.10 s. The average induced emf is 0.94 V.

Given :

A = 0.020 m²

N = 50

B = 0.18 T

t = 0.10 sec

Ф = 30° = ( π / 180 ) × 30 rad

Ф = 0.5236 rad

Now, angular velocity (ω) = Ф / t = 0.5236 / 0.10

ω = 5.236 rad/s

Average induced emf is given by

ε = N × A × B × ω

ε = 50 × 0.02 × 0.18 × 5.236

ε = 0.94 V

So, The average induced emf is 0.94 V.

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