AP Physics 1 2.8: Ideal Springs and Hooke’s Law

Describe the force exerted by an ideal spring and relate its magnitude and direction to displacement from the relaxed or equilibrium position.

[Maximum number: 4]

(12 points, suggested time 25 minutes)
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. All frictional forces are negligible.

Figure 1

Figure 2

(a)

At time $t=t_{1}$ , the rod is spinning such that the block moves in a circular path with a constant tangential speed $v_{1}$ and the spring is stretched a distance $d_{1}$ from the spring's unstretched length, as shown in Figure 1. At time $t=t_{2}$ , the rod is spinning such that the block moves in a circular path with a constant tangential speed $v_{2}$ and the spring is stretched a distance $d_{2}$ from the spring's unstretched length, where $d_{2}>d_{1}$ , as shown in Figure 2.

[ 3 ]

(i)

On the following dots, which represent the block at the locations shown in Figure 1 and Figure 2, draw the force that is exerted on the block by the spring at times $t=t_{1}$ and $t=t_{2}$ . The spring force must be represented by a distinct arrow starting on, and pointing away from, the dot.
Note: Draw the relative lengths of the vectors to reflect the relative magnitudes of the forces exerted by the spring at both times.

Figure for Question (i) — AP Physics 1: Algebra-Based

[ 2 ]

(i) For drawing rightward arrows in both diagrams 1 point
For the length of the arrow at $t=t_{2}$ being longer than the arrow at $t=t_{1}$ 1 point

Scoring Notes:

- A maximum of 1 point can be earned if extraneous unlabeled arrows are drawn.

- A maximum of 1 point can be earned if incorrect labeled forces are drawn.

Example Response

Figure for Answer to Question (i) — AP Physics 1: Algebra-Based

(ii)

Referencing $d_{1}$ and $d_{2}$ , describe your reasoning for drawing the arrows the length that you did in part (a)(i).

[ 1 ]

(ii) For an explanation that refers to the difference in the stretch length and indicates that the 1 point magnitude of the spring force is (or is not) related to the stretch length, consistent with the force diagram drawn in part (a)(i)
Example Response
The spring force arrow drawn at $t=t_{2}$ is longer because the spring is stretched a greater distance at that time and the spring force is related to the stretch distance.

(b)

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

[ 1 ]

(i)

Determine an expression for the magnitude of the net force $F_{\text {net }}$ exerted on the block. Express your answer in terms of $m_{0}, k_{0}, L, d$ , and fundamental constants, as appropriate.

[ 1 ]

(i) For the correct answer: $F_{\text {net }}=k_{0} d$

1 point

Scoring Notes:

- An answer of k x does not earn this point.

- Points for part (b)(i) may be earned if correct in (b)(ii).

Example Response

\begin{aligned} F_{\text {net }} & =\Sigma F=F_{s} \\ F_{\text {net }} & =\Sigma F=k_{0} d \end{aligned}

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2.8.A—Describe the force exerted on an object by an ideal spring question 1

Question (a)

Question (i)

Question (ii)

Question (b)

Question (i)