2024 Free Response · Set 1

Derive an expression for the speed $v$ of Block A at time $t_1$.

Derive an expression for the speed $v_{A,B}$ of the two-block system immediately after the collision at time $t_3$.

[Figure 1 repeated: same horizontal setup with Block A (mass $m$) against the spring, distance $x_c$, friction region of length $D$ with coefficient $\mu$ between $x_1$ and $x_2$, and Block B (mass $3m$) hanging by a string of length $\ell$ at $x_3$. Positions marked $0, x_0, x_1, x_2, x_3$ along axis $x$. Note: Figure not drawn to scale.]

On the following axes, sketch a graph of the kinetic energy $K$ of Block A as a function of time $t$ from time $t = 0$ to $t_3$.

[Blank graph: vertical axis $K$, horizontal axis $t$ with gridlines/tick marks at $t_1$, $t_2$, $t_3$ (origin at $0$); axes given with no curve drawn — student is to sketch the curve.]

Use principles of work and energy to justify the graph drawn in part (b)(i) for the time interval $t = 0$ to $t = t_1$. Explicitly reference features of the shape of the graph you drew in part (b)(i).

[Figure 2: The two-block system (combined mass $m + 3m$) hangs from a string of length $\ell$ attached to a fixed support at the top. The string makes an angle $\theta_{max}$ with the vertical (dashed vertical reference line shown), with the block system displaced to one side at time $t_4$, having swung from its position at time $t_3$ (shown hanging straight down). Note: Figure not drawn to scale.]

After the collision, the two-block system instantaneously comes to rest at time $t_4$, which occurs when the string makes a small angle $\theta_{max}$ with the vertical, as shown in Figure 2. For times $t > t_4$, the system oscillates with frequency $f_i$. The support holding the string is raised, and the procedure is then repeated using a new string of length $2\ell$.

Indicate how the new frequency of oscillation $f_{2\ell}$ of the system on the new string of length $2\ell$ will compare to the frequency of oscillation $f_i$ from the original procedure.

_____ $f_{2\ell} > f_i$ _____ $f_{2\ell} < f_i$ _____ $f_{2\ell} = f_i$

Briefly justify your answer.

Derive, but do NOT solve, a differential equation that could be used to determine the speed $v$ of the cylinder as a function of time $t$. Express your answer in terms of given quantities and physical constants, as appropriate.

[Figure 2: Graph of speed $v$ (vertical axis) versus time $t$ (horizontal axis), starting at origin $O$. The curve rises steeply at first, curving over and leveling off asymptotically at a horizontal line labeled $v_{max}$.]

The student correctly sketches the speed $v$ of the cylinder as a function of time $t$, as shown in Figure 2.

Draw a vertical line on the sketch in Figure 2 to indicate the earliest time at which $F_{drag}$ on the cylinder is equal to the magnitude of the weight of the cylinder. Label this time as $t_1$ on the time axis.

Justify the location of $t_1$. Explicitly reference appropriate features of the sketch in Figure 2.

Rather than dropping the cylinder from rest, the student throws the cylinder upward with a nonzero initial speed. The cylinder is in the same orientation as when the cylinder was previously dropped. The student allows the cylinder to fall toward the ground.

Indicate whether the magnitude of the cylinder's maximum downward speed after being thrown upward would be greater than, less than, or equal to the maximum speed $v_{max}$ in Figure 2.

_____ Greater than _____ Less than _____ Equal to

Briefly justify your answer.

The student conducts an experiment to better understand the relationship between maximum speed $v_{max}$ and mass. The student collects data to determine the maximum speed for cylinders dropped from rest, each with the same physical size and shape but a different mass $m$. The student then graphs $v_{max}^2$ as a function of mass.

[Scatter plot: vertical axis $v_{max}^2$ $(\text{m}^2/\text{s}^2)$ from $0$ to $10$ in increments of $2$; horizontal axis $m$ (kg) from $0.0$ to $0.6$ in increments of $0.1$. Data points approximately at: $(0.1, 2.6)$, $(0.15, 3.3)$, $(0.2, 4.0)$, $(0.3, 5.6)$, $(0.35, 6.0)$, $(0.45, 7.6)$, $(0.5, 7.5)$, $(0.55, 8.6)$. Gridlines shown at every $0.05$ kg horizontally and $0.5\ \text{m}^2/\text{s}^2$ vertically.]

Draw the best-fit line for the data.

Use the best-fit line to calculate an experimental value for $b$.

[Figure 3: A horizontal cylinder shown with several downward arrows above it (direction from which it was dropped) and an upward arrow labeled $F_{drag}$; a labeled dimension "Length" with an arrow spanning the cylinder's length.]

A student claims that the magnitude of the maximum speed of a cylinder dropped from rest depends on the length of the cylinder. The student designs an experiment to collect data that can be used to provide evidence to support the claim. The student drops cylinders with the orientation shown in Figure 3.

The student has access to but does not have to use all of the following equipment.

• Cylinder Set 1: cylinders of the same known length with different known masses
• Cylinder Set 2: cylinders of the same known mass with different known lengths
• A motion detector that can measure velocity as a function of time

Indicate two quantities that when graphed could be used to determine whether the length of the cylinder affects the maximum speed.

Vertical axis: _______________ Horizontal axis: _______________

Briefly describe how the quantities graphed could be used to determine the relationship between cylinder length and maximum speed.

[Figure: A schematic representation of the rod alone (no pole, no string, no block), extending diagonally with a pivot circle at the lower-left end. Points marked along the rod: P, C, and Q (in that order from the pivot outward).]

On the following representation of the rod, draw and label the forces (not components) that are exerted on the rod. Each force must be represented by a distinct arrow that starts on and points away from the point at which the force is exerted on the rod.

In Figure 1, Point P is located $\dfrac{3}{8}L$ from the pivot and Point Q is located $\dfrac{6}{8}L$ from the pivot.

Derive an equation for the tension $F_T$ in the horizontal string in terms of $L$, $m$, $\theta$, and physical constants, as appropriate.

[Figure 2: Same setup as Figure 1, but the string now connects from Point Q on the rod to a higher point on the vertical pole (the string attachment point is above the top of the pole shown in Figure 1), making a steeper angle with the pole. The rod still makes angle $\theta$ with the pole, with Points P, C, Q marked along the rod, and the block of mass $3m$ hanging from Point P. Pivot labeled "Pivot-$S$" at the base. Note: Figure not drawn to scale.]

The original string is replaced with a longer string that connects Point Q to a higher location on the vertical pole, as shown in Figure 2. The angle $\theta$ remains the same. How does the new tension $F_{T,\ new}$ compare with the original tension $F_T$ from part (b)? Justify your reasoning.

[Figure 3: A nonuniform rod attached horizontally to a pivot on a vertical pole, extending to the right along a horizontal $x$ (m) axis from $0$ at the pivot to $1.2$ at the far end labeled "Nonuniform Rod". A string labeled "String" connects from a point partway along the rod up to the top of the vertical pole, forming a right-triangle-like support. Pivot labeled "Pivot-$S$" at the base of the pole. Note: Figure not drawn to scale.]

A nonuniform rod is now attached to the pivot, as shown in Figure 3. There is negligible friction between the nonuniform rod and the pivot. The rod has a length of $1.2$ m and a linear mass density $\lambda(x) = A + Bx$, where $x$ is the distance from the pivot, $A = 6.0\ \text{kg/m}$, and $B = 10.0\ \text{kg/m}^2$.

Calculate the mass of the rod.

Calculate the rotational inertia of the rod about the pivot.