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AP®PhysicsCollege BoardPhysics 1: Algebra-BasedExam QuestionsUnit 6: Energy & Momentum of Rotating SystemsExamples of Rotational Systems

Examples of Rotational Systems (College Board AP® Physics 1: Algebra-Based): Exam Questions

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Examples of Rotational Systems

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1a
Sme Calculator7 marks
A disk with mass M and radius R rolling down a ramp inclined at angle θ, with motion lines indicating the disk is moving down the ramp.

Figure 1

In Experiment 1, students release a disk of mass M and radius R from rest at the top of a ramp that makes an angle theta with the horizontal. The disk rolls without slipping, as shown in Figure 1. The rotational inertia of the disk is I subscript d i s k end subscript space equals space 1 half M R squared. The frictional force exerted on the disk is F subscript f.

i) On the diagram in Figure 2, draw and label arrows that represent the forces (not components) that are exerted on the disk as it rolls down the ramp, which is indicated by the dashed line. Each force in your diagram must be represented by a distinct arrow starting on, and pointing away from, the point at which the force is exerted on the disk.

A blank force diagram of the disk (circle) on the ramp (slanted dashed line).

Figure 2

ii) Determine an expression for the net torque exerted on the disk around the center in terms of theta, F subscript f, M, R and physical constants as appropriate.

iii) Determine an expression for the net force exerted on the disk in terms of theta, F subscript f, M, R and physical constants as appropriate.

iv) Derive an expression for the translational acceleration of the center of mass of the disk. Express your answer in terms of M, theta, R and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

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1b
Sme Calculator3 marks
In Experiment 1, a sphere rolls down a slope, and in Experiment 2 an ice cube slides down an identical slope, both at angle θ.

Figure 3

In Experiment 1, the disk reaches the bottom of the ramp in time t subscript d i s k end subscript. In Experiment 2, the students release a block of ice from rest from the same height on a ramp inclined at the same angle theta, as shown in Figure 3. Friction between the block and the ramp is negligible. The block reaches the bottom of the ramp in time t subscript b l o c k end subscript.

Indicate whether t subscript d i s k end subscript is greater than, less than, or equal to t subscript b l o c k end subscript.

⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ t subscript d i s k end subscript space greater than space t subscript b l o c k end subscript‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ t subscript d i s k end subscript space equals space t subscript b l o c k end subscript ‎ ‎ ‎‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ t subscript d i s k end subscript space less than space t subscript b l o c k end subscript

Justify your reasoning.

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2
Sme Calculator2 marks
A spacecraft orbiting the Earth in a circular path of radius R from the Earth's center.

Figure 1

A spacecraft of mass m is in a clockwise circular orbit of radius R around Earth, as shown in Figure 1. The mass of Earth is M subscript E.

The spacecraft is moved into a new circular orbit that has a radius greater than R, as shown in Figure 2.

A spacecraft changing from an orbital path of radius R to a new orbit with a radius greater than R.

Figure 2

Is the total mechanical energy of the spacecraft in the new orbit greater than, less than, or equal to the total mechanical energy of the spacecraft in the original orbit? Briefly justify your reasoning.

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3
Sme Calculator2 marks
A ball on the ground just below a pivot connected to a horizontal rod, with a curved dashed line representing the rotational path of the end of the rod once released. To the left of the ball, points A and B are marked on the horizontal surface which is shaded grey to indicate the surface is rough.

Figure 1

A system consists of a small sphere of mass m and radius R at rest on a horizontal surface and a uniform rod of mass M space equals space 2 m and length l attached at one end to a pivot with negligible friction, where R space much less-than space l. There is negligible friction between the surface and the sphere to the right of Point A and non-negligible friction to the left of Point A. The rod is held horizontally as shown in Figure 1, and then is released from rest. The total rotational inertia of the rod about the pivot is 1 third M l squared and the rotational inertia of the sphere about its center is 2 over 5 m R squared. After the rod is released, the rod swings down and strikes the sphere head-on. As a result of this collision, the rod is stopped, and the ball initially slides without rolling to the left across the horizontal surface.

When the sphere encounters Point A, it begins rolling while slipping, and eventually begins rolling without slipping at Point B.

The circle in Figure 2 represents the sphere while it is traveling between Points A and B. On the diagram in Figure 2, draw and label the forces (not components) that act on the sphere. Each force must be represented by a distinct arrow starting on, and pointing away from, the point of application on the sphere.

A circle to be used to represent the forces acting on the sphere as it rolls while slipping.

Figure 2

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4a1 mark
An elliptical orbit around a planet with points A, B, and C marked on the ellipse. The planet is to the left of the center of the ellipse, point A is to the left of the planet and point C is to the right. Point B is above the center of the ellipse. Arrows indicate the direction of orbit is clockwise.

Figure 1

A satellite is moving clockwise in an elliptical orbit around a planet, as shown in Figure 1. The satellite is closest to the planet at Point A and furthest at Point C. Point B is equidistant from points A and C. The distances from the planet to points A, B, and C are r subscript A, r subscript B, and r subscript C, respectively.

Figures 2 and 3 show the planet and part of the satellite's orbit, including points A and B.

Two identical diagrams labeled Figure 2 and Figure 3, showing a portion of the elliptical orbit with the planet, point A, and point B.

i) On the diagram in Figure 2, draw arrows to indicate the direction of the satellite's velocity at points A and B. If the velocity is zero at either point, write “v space equals space 0” next to the appropriate dot.

ii) On the diagram in Figure 3, draw arrows to indicate the direction of the satellite's acceleration at points A and B. If the acceleration is zero at either point, write “a space equals space 0” next to the appropriate dot.

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4b1 mark
Elliptical orbit showing the distance from A to the planet is rA, the distance from B to the planet is 2.5 times rA, and the acute angle between the lines is 53 degrees.

Figure 4

The mass of the satellite is m subscript S. At Point A, its speed is v subscript A and it is a distance r subscript A from the planet. At Point B, its speed is v subscript B and it is a distance r subscript B space equals space 5 over 2 r subscript A from the planet, as shown in Figure 4. The angle between the lines joining the planet and Point B and the planet and Point C is 53 degree.

Starting with conservation of angular momentum, derive an expression for the speed v subscript B of the satellite at Point B. Express your answer in terms of m subscript S, v subscript A, r subscript A, and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

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4c1 mark

On the axes in Figure 5, sketch a graph of the satellite's kinetic energy as a function of distance r from the planet.

Graph axes showing kinetic energy versus distance r, with points at KA, 2KA, rA, and rC, and dashed lines connecting these points horizontally and vertically.

Figure 5

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4d1 mark

Indicate whether the graph you drew in part c) is consistent with the velocity and acceleration you indicated for Point B in part a). Justify your answer.

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51 mark
The Earth with radius R, with a dotted line around it showing an orbit of radius 2R. Two objects, mass m and 3m, move at velocity v0 along the orbital path towards each other.

Figure 1

Two satellites, of masses m and 3 m, respectively, are in the same circular orbit about the Earth’s center, as shown in Figure 1. The Earth has mass M and radius R. In this orbit, which has a radius of 2 R, the satellites initially move with the same orbital speed v subscript 0 but in opposite directions. The satellites collide head-on and stick together. After the collision, the satellites move with orbital speed v.

i) Determine an expression for the orbital speed v subscript 0 of the satellites before the collision in terms of M, R, and physical constants as appropriate.

ii) Determine an expression for the speed v of the satellites immediately after the collision, in terms of v subscript 0.

iii) Derive an expression for the total mechanical energy of the system immediately after the collision in terms of m, M, R, and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

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1a
Sme Calculator3 marks
A ramp with a hollow sphere, a solid sphere, and a hoop at the top. Incline angle is θ and ramp length is L.

Figure 1

A hollow sphere, a uniform solid sphere, and a hoop are placed at the top of a ramp of length L that makes an angle theta with the horizontal, as shown in Figure 1. The hollow sphere, solid sphere, and hoop have the same mass M and the same radius R. All three shapes are released from rest at the same time and roll down the ramp without slipping.

Rank the order in which the shapes reach the bottom of the ramp. A ranking of “1” indicates the shape reaches the bottom of the ramp first. If any two shapes reach the bottom of the ramp at the same time, give them the same ranking.

⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ Hollow Sphere‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ Solid Sphere ‎ ‎ ‎‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ Hoop

Justify your rankings using relevant physics principles, but without directly manipulating or deriving equations.

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1b
Sme Calculator3 marks

The time for one of the shapes to roll without slipping to the bottom of the ramp is t, and the rotational inertia of that shape is I.

Derive the relationship between t and I to show that t space equals space square root of fraction numerator 2 L open parentheses I space plus space M R squared close parentheses over denominator M g R squared space sin space theta end fraction end root. Begin your derivation by writing a fundamental physics principle or an equation from the reference material.

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1c
Sme Calculator2 marks

A new sphere is constructed using a thin spherical shell of radius R. The shell is completely filled with liquid. The mass of the shell is much less than that of the liquid, and the total mass of the liquid-filled sphere is M. The liquid-filled sphere is placed at the top of the ramp, released from rest, and rolls without slipping to the bottom of the ramp. As the sphere rolls down the ramp, the liquid inside the sphere does not rotate with the shell.

Does the liquid-filled sphere reach the bottom of the ramp in more time, less time, or the same time as the solid sphere? Use the equation derived in part b) to justify your answer.

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