Spring Forces (College Board AP® Physics 1: Algebra-Based): Exam Questions

Figure 1

Two ideal springs, 1 and 2, of spring constant $k_1$ and $k_2$ respectively, are connected end to end. A block of mass $m$ is attached to the end of Spring 2 and the other end of Spring 1 is fixed to a wall. The block is displaced to the left of the spring's equilibrium position, $x=0$, and held stationary at position $x=-A$, as shown in Figure 1.

On the diagram in Figure 1, draw and label arrows that represent the forces (not components) that are exerted on Spring 2 as the block is held at position $x=-A$.

• 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 Spring 2.

• The length of arrows should represent the relative magnitudes of the forces.

Figure 1

A group of students is asked to take measurements to create a graph that can be used to determine the mass $m_B$ of a block. The students have a set of springs that are attached at one end to the ceiling. Each spring has a different, known spring constant and a loop that the block can be attached to. The students are also given a meterstick but do not have access to a scale or balance.

i) Indicate quantities that could be measured by the students that would allow them to determine the mass $m_B$ of the block using a linear graph.

ii) Briefly describe a method to reduce experimental uncertainty for the measured quantities.

i) Indicate what quantities students could graph on the horizontal and vertical axes to create a linear graph that can be used to determine the mass $m_B$ of the block.

ii) Briefly describe the relationship between the mass $m_B$ of the block and a feature of the graph from part b)i). Your answer may include an equation that relates the mass and the chosen feature of the graph.

Figure 1

A small block of mass $m_0$ is attached to the end of a spring of spring constant $k_0$ that is attached to a rod on a horizontal table. The rod is attached to a motor so that the rod can rotate at various speeds about its axis. When the rod is not rotating, the block is at rest and the spring is at its unstretched length $L$, as shown in Figure 1. All frictional forces are negligible.

At time $t=t_1$, the rod is spinning such that the block moves in a circular path with constant tangential speed $v_1$ and the spring is stretched a distance $d_1$ from its unstretched length, as shown in Figure 2. At time $t=t_2$, the rod is spinning such that the block moves in a circular path with constant tangential speed $v_2$ and the spring is stretched a distance $d_2$ from its unstretched length, where $d_2>d_1$, as shown in Figure 3.

Indicate whether the tangential speed $v_1$ of the block at time $t=t_1$ is greater than, less than, or equal to the tangential speed $v_2$ of the block at time $t=t_2$ by writing one of the following:

• $v_1>v_2$

• $v_1<v_2$

• $v_1=v_2$

Justify your answer. In your justification, include qualitative reasoning beyond mathematical derivations or expressions.

Consider a scenario where the block travels in a circular path where the spring is stretched a distance $d$ from its unstretched length $L$.

Derive an equation for the tangential speed $v$ of the block. Express your answer in terms of $m_0$, $k_0$, $L$, $d$, and fundamental constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Does the equation you derived in part b) agree with your qualitative reasoning from part a)? Justify why or why not.

Figure 1

A block of mass $m_B$ is attached to an ideal spring, whose other end is fixed to the ceiling, as shown in Figure 1. The block is displaced a distance $y_0$ below the spring’s equilibrium position. The block is then released from rest and oscillates vertically.

The dots in Figure 2 represent the block as it passes through positions $y=0$ and $y=+y_0$ while the block oscillates. On the dots in Figure 2, draw and label the forces (not components) that are exerted on the block at positions $y=0$ and $y=+y_0$. Each force must be represented by a distinct arrow starting on and pointing away from the appropriate dot. The lengths of the arrows should reflect the relative magnitudes of the forces.

Figure 2