AP Physics C Mechanics Quiz: Spring Forces

A force $F$ is required to stretch a single ideal spring with constant $k$ by a distance $x$. If two identical springs are connected in parallel, what total force is required to stretch the combination by the same distance $x$?

An ideal spring with spring constant $k$ is cut into two identical halves. One of the halves is designated Spring A. The two halves are then connected in parallel to form a new spring system, designated System B. What is the ratio of the spring constant of System B to the spring constant of Spring A?

An ideal spring is fixed at one end to a wall. When a block attached to the other end is at position $x > 0$, the spring is stretched. When the block is at $x < 0$, the spring is compressed. The equilibrium position is at $x=0$. Which statement correctly describes the force $\vec{F}_s$ exerted by the spring on the block?

A non-ideal spring exerts a restoring force on an object given by $F = -ax - bx^3$, where $a$ and $b$ are positive constants. The object is displaced from equilibrium by a small distance $x_0$. For this displacement, the contribution from the cubic term is negligible. If the displacement is doubled to $2x_0$ such that the cubic term is no longer negligible, how does the magnitude of the restoring force compare to the ideal spring force $F_{ideal} = k(2x_0)$ with $k=a$?

A block is placed on a frictionless horizontal surface between two walls. Two ideal springs, with constants $k_1$ and $k_2$, are connected to opposite sides of the block and to the walls. The springs are at their natural lengths at the equilibrium position $x=0$. If the block is displaced a distance $x$ from equilibrium, what is the effective spring constant $k_{\text{eff}}$ of the system that provides the net restoring force?

A student has two ideal springs, A and B, with spring constants satisfying $k_A > k_B$. The student connects them in series, hangs a block from the combination, and measures the stretch. The student then connects them in parallel, hangs the same block, and measures the stretch. The student claims that the stretch will be greater in the series configuration. Which of the following correctly evaluates this claim?

A $0.250\ \text{kg}$ block is attached to two ideal springs on a horizontal frictionless table. Spring 1 has constant $k_1 = 120\ \text{N/m}$ and Spring 2 has constant $k_2 = 80.0\ \text{N/m}$. The springs are connected in parallel between a wall and the block so that both springs stretch or compress by the same amount x when the block is displaced. The block is pulled a small distance and released, oscillating about equilibrium.

Given values:

• $k_1 = 120\ \text{N/m}$
• $k_2 = 80.0\ \text{N/m}$
• $m = 0.250\ \text{kg}$

Forces and model: For a displacement x, each spring exerts a restoring force $F_{s1} = -k_1 x$ and $F_{s2} = -k_2 x$. The net restoring force is $F_{\text{net}} = -(k_1 + k_2)x,$ so Newton's Second Law becomes $m\,\frac{d^2x}{dt^2} = -(k_1+k_2)x.$

Refer to the system described above. What is the effective spring constant for the system?

A $2.0\ \text{kg}$ cart is attached to two springs connected in series along a horizontal frictionless track. Spring A has constant $k_A = 300\ \text{N/m}$ and spring B has constant $k_B = 600\ \text{N/m}$. The left end of spring A is fixed to a wall, spring A connects to spring B, and spring B connects to the cart. When the cart is displaced to the right from equilibrium by a small distance, both springs stretch, and each spring exerts a restoring force. For springs in series, the same force magnitude acts through both springs, and the total extension is the sum of individual extensions. The effective spring constant satisfies $\tfrac{1}{k_{\text{eff}}} = \tfrac{1}{k_A} + \tfrac{1}{k_B}$. The cart oscillates with small amplitude so that Hooke's law applies.

Given values:

• Mass: m = $2.0\ \text{kg}$
• Spring constants: $k_A = 300\ \text{N/m}$, $k_B = 600\ \text{N/m}$
• Track is horizontal and frictionless

Refer to the system described above. What is the effective spring constant for the system?

A $0.600\ \text{kg}$ block is connected to two ideal springs in series on a frictionless horizontal surface. Spring 1 has constant $k_1 = 300\ \text{N/m}$ and Spring 2 has constant $k_2 = 150\ \text{N/m}$. The left end of Spring 1 is attached to a wall, Spring 1 connects to Spring 2, and the right end of Spring 2 attaches to the block. The block is displaced slightly and released, oscillating with small amplitude.

Given values:

• $k_1 = 300\ \text{N/m}$
• $k_2 = 150\ \text{N/m}$
• $m = 0.600\ \text{kg}$

Forces and model: In series, both springs carry the same force magnitude F while their extensions add: $x = x_1 + x_2$. Hooke's Law gives $F = k_1 x_1 = k_2 x_2$, so the equivalent spring constant satisfies $\frac{1}{k_{\text{eff}}} = \frac{1}{k_1} + \frac{1}{k_2}.$ The motion then satisfies $m\,d^2x/dt^2 = -k_{\text{eff}}x$.

Refer to the system described above. What is the effective spring constant for the system?

A $0.50\ \text{kg}$ cart on a frictionless horizontal track is attached between two springs: spring 1 on the left with constant $k_1 = 120\ \text{N/m}$ and spring 2 on the right with constant $k_2 = 80\ \text{N/m}$. Each spring is fixed to a wall at its far end, and the cart is connected to both springs so that when the cart is displaced a distance x to the right from equilibrium, the left spring stretches by x and the right spring compresses by x. Both springs obey Hooke's law, and the restoring forces add. Along the track, the only forces on the cart are the spring forces; weight and normal cancel vertically. For a displacement x, the net restoring force is $F_{\text{net}} = -(k_1 + k_2)x$, consistent with Newton's second law $\sum F = ma$ for simple harmonic motion.

Given values:

• Mass: m = $0.50\ \text{kg}$
• Spring constants: $k_1 = 120\ \text{N/m}$, $k_2 = 80\ \text{N/m}$
• Motion is horizontal and frictionless

Refer to the system described above. What is the effective spring constant for the system?

An ideal horizontal spring with a spring constant of $k = 50 \, \text{N/m}$ is attached to a block on a frictionless surface. The block is pulled, stretching the spring by $0.20 \, \text{m}$ from its equilibrium position. What is the magnitude of the restoring force exerted by the spring on the block?

A block of mass $m = 2.0 \, \text{kg}$ is suspended from a vertical ideal spring, causing the spring to stretch by $0.10 \, \text{m}$ from its natural length once it reaches equilibrium. What is the spring constant $k$ of the spring? (Use $g \approx 10 \, \text{m/s}^2$)

Two ideal springs with spring constants $k_1 = 100 \, \text{N/m}$ and $k_2 = 300 \, \text{N/m}$ are attached in parallel to a block on a frictionless horizontal surface. What is the equivalent spring constant $k_{\text{eq}}$ of this spring system?

Two ideal springs with spring constants $k_1 = 100 \, \text{N/m}$ and $k_2 = 300 \, \text{N/m}$ are connected in series. What is the equivalent spring constant $k_{\text{eq}}$ of this combination?

A block of mass $M$ rests on a frictionless plane inclined at an angle $\theta$ to the horizontal. The block is attached to an ideal spring with spring constant $k$, which is fixed to the top of the incline. The spring is stretched by a distance $x$ from its natural length to hold the block in equilibrium. What is the expression for $x$?

A block of mass $m$ is attached to a horizontal ideal spring of constant $k$. The block is displaced a distance $A$ from its equilibrium position $x=0$ and released from rest on a frictionless surface. What is the magnitude of the initial acceleration of the block at the moment of release?

The restoring force exerted by an ideal spring has magnitude $F$ when it is stretched a distance $x$ from its equilibrium position. If the spring is instead stretched by a distance $3x$, what is the new magnitude of the restoring force?

A mass $m$ is hung from an ideal spring of constant $k$, causing it to stretch a distance $x$. A second, identical spring is then attached in series with the first, and the same mass $m$ is hung from the combination. What is the total stretch of the two-spring combination?

The magnitude of the force $F$ required to stretch an ideal spring is measured as a function of the spring's elongation $x$. A graph of $F$ versus $x$ is created, which is a straight line passing through the origin. The slope of this line represents which physical quantity?

An ideal spring with spring constant $k$ is initially compressed by a distance $x_0$ from its equilibrium length. An external force is applied to compress it further to a final compression of $3x_0$. What is the change in the magnitude of the force exerted by the spring?