AP Physics C Mechanics Unit 2: Force and Translational Dynamics

Explain translational motion through system models, center of mass, force diagrams, Newton’s laws, contact forces, resistive forces, springs, and circular motion.

Covered syllabus points

2.1 Systems and Center of Mass 2.2 Forces and Free-Body Diagrams 2.3 Newton’s Third Law 2.4 Newton’s First Law 2.5 Newton’s Second Law 2.6 Gravitational Force 2.7 Kinetic and Static Friction 2.8 Spring Forces 2.9 Resistive Forces 2.10 Circular Motion

[Maximum number: 2]

Cart 1 of mass $m_{1}$ is held at rest above the bottom of an incline. Cart 2 has mass $m_{2}$ , where $m_{2}>m_{1}$ , and is at rest at the bottom of the incline. At time t=0, Cart 1 is released and then travels down the incline and smoothly transitions to the horizontal section. The center of mass of Cart 1 moves a vertical distance of H, as shown. At time $t_{\mathrm{C}}$ , Cart 1 reaches the bottom of the incline and immediately collides with and sticks to Cart 2. After the collision, the two-cart system moves with constant speed v. Frictional and rotational effects are negligible.

During the collision, is the impulse on Cart 1 from Cart 2 greater than, less than, or equal to the magnitude of the impulse on Cart 2 from Cart 1 ?

Greater than Less than Equal to

Justify your answer.

[Maximum number: 9]

Blocks of mass m and 2 m are connected by a light string and placed on a frictionless inclined plane that makes an angle $\theta$ with the horizontal, as shown in Figure 1 above. Another light string connecting the block of mass m to a hanging sphere of mass M passes over a pulley of negligible mass and negligible friction. The entire system is initially at rest and in equilibrium.

(a)

On the dots below that represent the block of mass m and the sphere of mass M, draw and label the forces (not components) that act on each of the objects shown. Each force must be represented by a distinct arrow starting on and pointing away from the dot.

Figure for Question (a) — AP Physics C: Mechanics

[ 3 ]

(b)

Derive expressions for the magnitude of each of the following. If you need to draw anything other than what you have shown in part (a) to assist in your solution, use the space below. Do NOT add anything to the figures in part (a).

The force $T_{2}$ exerted on the block of mass m by the string. Express your answers in terms of $m, \theta$ , and physical constants, as appropriate.

The mass M for which the system can remain in equilibrium. Express your answers in terms of $m, \theta$ , and physical constants, as appropriate.

[ 2 ]

(c)

Now suppose that mass M is large enough to descend and that the sphere reaches the floor before the blocks reach the pulley. Answer the following for the moment immediately after the sphere reaches the floor.

Does the tension $T_{1}$ increase, decrease to a nonzero value, decrease to zero, or stay the same? Increase Decrease to a nonzero value Decrease to zero Stay the same

Is the velocity of the block of mass m up the ramp, down the ramp, or zero? Up the ramp Down the ramp Zero

Is the acceleration of the block of mass m up the ramp, down the ramp, or zero? Up the ramp Down the ramp Zero

[ 1 ]

(d)

Consider the initial setup in Figure 1. Now suppose the surface of the incline is rough and the coefficient of static friction between the blocks and the inclined plane is $\mu_{s}$ . Derive an expression for the minimum possible value of M that will keep the blocks from moving down the incline. Express your answer in terms of $m, \mu_{s}, \theta$ , and fundamental constants, as appropriate.

[ 3 ]

[Maximum number: 3]

Blocks A and B of masses 2 m and m, respectively, are arranged in a setup consisting of a ramp that makes an angle $\theta$ with a smooth horizontal table and an ideal spring of spring constant k fixed to a wall, as shown. Block A is held at rest a distance D up the ramp, and Block B is at rest on the horizontal table. The coefficient of kinetic friction between Block A and the rough ramp is $\mu$ in the region of length D, and there is negligible friction between the blocks and the smooth table.

At time t=0, Block A is located at horizontal position x=0 and is released from rest. After the block is released, the following occurs.

- At time $t=t_{1}$ , Block A has traveled a distance D down the ramp, has transitioned to the table, and is moving with speed v at $x=x_{1}$ .

- At time $t=t_{2}$ , Block A is at $x=x_{2}$ when it collides with and sticks to Block B.

- At time $t=t_{3}$ , the combined blocks A and B are at $x=x_{3}$ when they collide with and stick to the spring in its equilibrium position.

- At time $t=t_{4}$ , the combined blocks A and B are instantaneously at rest and the spring is compressed a distance $x_{\mathrm{c}}$ from its equilibrium position.

Use principles of forces to justify the graph drawn in part (b)(i) for the time interval $t=t_{3}$ to $t=t_{4}$ . Explicitly reference features of the shape of the graph you drew in part (b)(i).

For times $t>t_{4}$ , the two-block-spring system oscillates with period $T_{\mathrm{O}}$ . The procedure is then repeated using a new ramp, where there is negligible friction between Block A and the ramp.

(ii) For a statement about the change in momentum that is consistent with the graph drawn in the response for part (b)(i)
For indicating that a decreasing graph means that the force exerted on Block A is in a direction opposite to the motion of Block A Scoring Note: A response that indicates that an increasing graph means that the force exerted on Block A is in the same direction as the motion of Block A also earns this point.
For relating the change in momentum to the magnitude of the force exerted on Block A 1 point
Example Response
The momentum of Block A decreases between $t_{3}$ and $t_{4}$ because the spring exerts a force on the blocks in the opposite direction of the velocity of the blocks, causing the blocks to slow to a The spring force increases the more the spring compresses, so the momentum of Block A decreases at an increasing rate, which is shown in the slope of the curve becoming steeper with time.
Total for part (b) 7 points

[Maximum number: 1]

Two moons of equal mass are in circular orbits around a planet. Moon 1 orbits with a radius of r and moon 2 orbits with a radius of 3 r. What is the ratio of the magnitude of the planet's gravitational force on moon 2 to that on moon 1, $F_{G(\text { on } 2)} / F_{G(\text { on } 1)}$ ?

9/1

3/1

1/1

1/3

1/9

All question bank results loaded

Covered syllabus points

2 Force and Translational Dynamics question 1

2 Force and Translational Dynamics question 2

Question (a)

Question (b)

Question (c)

Question (d)

2 Force and Translational Dynamics question 3

2 Force and Translational Dynamics question 4