Learn Extracted exam questions AP Physics C: Mechanics 2017 Free Response
2017 Free Response
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An Atwood's machine consists of two blocks connected by a light string that passes over a frictionless pulley of negligible mass, as shown in the figure above. The masses of the two blocks, $M_1$ and $M_2$, can be varied. $M_2$ is always greater than $M_1$.
[Figure: An Atwood's machine — two blocks $M_1$ and $M_2$ hang from a light string over a frictionless pulley of negligible mass mounted on a support stand, above a table/floor surface.]
On the dots below, which represent the blocks, draw and label the forces (not components) that act on the blocks. Each force must be represented by a distinct arrow starting on and pointing away from the appropriate dot. The relative lengths of the arrows should show the relative magnitudes of the forces.
[Two dotted grids, each with a single large dot at its center, labeled "Block of Mass $M_1$" and "Block of Mass $M_2$" respectively — students draw force vectors from each dot.]
Using the forces in your diagrams above, write an equation applying Newton's second law to each block and use these two equations to derive the magnitude of the acceleration of the blocks and show that it is given by the equation:
The magnitude of the acceleration $a$ was measured for different values of $M_1$ and $M_2$, and the data are shown below.
| $M_1$ (kg) | 1.0 | 2.0 | 5.0 | 6.0 | 10.0 |
|---|---|---|---|---|---|
| $M_2$ (kg) | 2.0 | 3.0 | 12.0 | 8.0 | 14.0 |
| $a$ (m/s$^2$) | 3.02 | 1.82 | 4.21 | 1.15 | 1.71 |
Indicate below which quantities should be graphed to yield a straight line whose slope could be used to calculate a numerical value for the acceleration due to gravity $g$.
Vertical axis: ____________________
Horizontal axis: ____________________
Use the remaining rows in the table above, as needed, to record any quantities that you indicated that are not given.
Plot the data points for the quantities indicated in part (c) on the graph below. Clearly scale and label all axes including units, if appropriate. Draw a straight line that best represents the data.
[Blank grid for plotting the linearized data from part (c), with a best-fit straight line to be drawn through the plotted points.]
Using your straight line, determine an experimental value for $g$.
[Figure: The experiment is now modified — block $M_1$ rests on a smooth, horizontal table, with the string passing over a frictionless pulley mounted at the edge of the table, and block $M_2$ hangs vertically over the side of the table.]
The experiment is now repeated with a modification. The Atwood's machine is now set up so that the block of mass $M_1$ is on a smooth, horizontal table and the block of mass $M_2$ is hanging over the side of the table, as shown in the figure above.
For the same values of $M_1$ and $M_2$, is the magnitude of the tension in the string when the blocks are moving higher, lower, or equal to the magnitude of the tension in the string when the blocks are moving in the first experiment?
____ Higher ____ Lower ____ Equal to
Justify your answer.
The value determined for the acceleration due to gravity $g$ is lower than in the first experiment. Give one physical factor that could account for this lower value and explain how this factor affected the experiment.
[Figure: A block of mass $m$ rests at the top of an inclined plane of height $h$, which descends smoothly onto a horizontal surface. Farther along the horizontal surface, a coiled spring is fixed against a wall. Note: Figure not drawn to scale.]
A block of mass $m$ starts at rest at the top of an inclined plane of height $h$, as shown in the figure above. The block travels down the inclined plane and makes a smooth transition onto a horizontal surface. While traveling on the horizontal surface, the block collides with and attaches to an ideal spring of spring constant $k$. There is negligible friction between the block and both the inclined plane and the horizontal surface, and the spring has negligible mass. Express all algebraic answers for parts (a), (b), and (c) in terms of $m$, $h$, $k$, and physical constants, as appropriate.
Derive an expression for the speed of the block just before it collides with the spring.
Is the speed halfway down the incline greater than, less than, or equal to one-half the speed at the bottom of the inclined plane?
____ Greater than ____ Less than ____ Equal to
Justify your answer.
Derive an expression for the maximum compression of the spring.
Determine an expression for the time from when the block collides with the spring to when the spring reaches its maximum compression.
The block is again released from rest at the top of the incline, and when it reaches the horizontal surface it is moving with speed $v_0$. Now suppose the block experiences a resistive force as it slides on the horizontal surface.
The magnitude of the resistive force $F$ is given as a function of speed $v$ by $F = \beta v^2$, where $\beta$ is a positive constant with units of $\text{kg/m}$.
Write, but do NOT solve, a differential equation for the speed of the block on the horizontal surface as a function of time $t$ before it reaches the spring. Express your answer in terms of $m$, $h$, $k$, $\beta$, $v$, and physical constants, as appropriate.
Using the differential equation from part (d)i, show that the speed of the block $v(t)$ as a function of time $t$ can be written in the form $\dfrac{1}{v(t)} = \dfrac{1}{v_0} + \dfrac{\beta t}{m}$, where $v_0$ is the speed at $t = 0$.
Sketch graphs of position $x$ as a function of time $t$, velocity $v$ as a function of time $t$, and acceleration $a$ as a function of time $t$ for the block as it is moving on the horizontal surface before it reaches the spring.
[Three blank axes side by side, each with a vertical axis and horizontal axis labeled $t$: the first labeled $x$, the second labeled $v$, the third labeled $a$ — for sketching the respective graphs.]
[Figure: A uniform solid cylinder of mass $M = 0.50$ kg and radius $R = 0.10$ m sits at the top of a 1.0 m long inclined plane making a $30°$ angle with the horizontal. The incline descends onto a table of height 0.75 m. The cylinder rolls down the incline, across the table, and launches horizontally off the edge of the table, following a projectile path (shown dashed) to land on the floor a horizontal distance $D$ from the edge of the table. A coiled spring/bumper is shown mounted at the far right on the floor.]
A uniform solid cylinder of mass $M = 0.50$ kg and radius $R = 0.10$ m is released from rest, rolls without slipping down a 1.0 m long inclined plane, and is launched horizontally from a horizontal table of height 0.75 m. The inclined plane makes an angle of $30°$ with the horizontal. The cylinder lands on the floor a distance $D$ away from the edge of the table, as shown in the figure above. There is a smooth transition from the inclined plane to the horizontal table, and the motion occurs with no frictional energy losses. The rotational inertia of a cylinder around its center is $MR^2/2$.
Calculate the total kinetic energy of the cylinder as it reaches the horizontal table.
Calculate the angular velocity of the cylinder around its axis at the moment it reaches the floor.
Calculate the ratio of the rotational kinetic energy to the total kinetic energy for the cylinder at the moment it reaches the floor.
Calculate the horizontal distance $D$.
A sphere of the same mass and radius is now rolled down the same inclined plane. The rotational inertia of a sphere around its center is $\dfrac{2}{5}MR^2$.
Is the total kinetic energy of the sphere at the moment it reaches the floor greater than, less than, or equal to the total kinetic energy of the cylinder at the moment it reaches the floor?
____ Greater than ____ Less than ____ Equal to
Justify your answer.
Is the rotational kinetic energy of the sphere at the moment it reaches the floor greater than, less than, or equal to the rotational kinetic energy of the cylinder at the moment it reaches the floor?
____ Greater than ____ Less than ____ Equal to
Justify your answer.
Is the horizontal distance the sphere travels from the table to where it hits the floor greater than, less than, or equal to the horizontal distance the cylinder travels from the table to where it hits the floor?
____ Greater than ____ Less than ____ Equal to
Justify your answer.