AP Physics 1: Work Done by a Variable Force Quiz with Answer Key

📚 Quick Study Guide

• 📈 Work Done by a Variable Force: When the force acting on an object is not constant, the work done cannot be calculated using $W = Fd\cos(\theta)$. Instead, we must use integration.
• ➗ One Dimension: In one dimension, the work done by a variable force $F(x)$ from position $x_i$ to $x_f$ is given by: $W = \int_{x_i}^{x_f} F(x) dx$. This represents the area under the force vs. position curve.
• 📐 Two or Three Dimensions: In multiple dimensions, the work done is given by the line integral: $W = \int_{C} \vec{F} \cdot d\vec{r}$, where $\vec{F}$ is the force vector and $d\vec{r}$ is the infinitesimal displacement vector along the path $C$.
• 🌱 Spring Force: The force exerted by a spring is given by Hooke's Law: $F(x) = -kx$, where $k$ is the spring constant and $x$ is the displacement from the equilibrium position. The work done to stretch or compress a spring is $W = \frac{1}{2}kx^2$.
• ⚡ Graphical Interpretation: The work done by a variable force can be graphically determined by finding the area under the force vs. displacement curve.

Practice Quiz

1. A force acting on an object is given by $F(x) = 3x^2 - 2x$, where $F$ is in Newtons and $x$ is in meters. What is the work done by this force as the object moves from $x = 1$ m to $x = 3$ m?

   1. A) 16 J
   2. B) 20 J
   3. C) 22 J
   4. D) 24 J

2. The force exerted by a spring is given by $F(x) = -50x$ N. How much work is required to stretch the spring from its equilibrium position to $x = 0.2$ m?

   1. A) 1 J
   2. B) 2 J
   3. C) 3 J
   4. D) 4 J

3. A particle moves along the x-axis under the influence of a force $F(x) = \frac{C}{x^2}$, where $C$ is a constant. What is the work done by this force when the particle moves from $x = a$ to $x = 3a$?

   1. A) $\frac{2C}{3a}$
   2. B) $\frac{C}{3a}$
   3. C) $\frac{4C}{3a}$
   4. D) $\frac{5C}{3a}$

4. A force $F(x) = 4x^3$ (in Newtons) acts on an object. Find the work done by the force in moving the object from $x = 0$ to $x = 2$ meters.

   1. A) 4 J
   2. B) 8 J
   3. C) 16 J
   4. D) 32 J

5. The graph of force versus position for a variable force is a triangle with a base of 4 meters and a height of 10 N. How much work does the force do over these 4 meters?

   1. A) 10 J
   2. B) 20 J
   3. C) 30 J
   4. D) 40 J

6. A force is given by $\vec{F} = (2x\hat{i} + 3y^2\hat{j})$ N. Calculate the work done by this force in moving an object from (0, 0) to (2, 1) along a straight line path, where x and y are in meters.

   1. A) 4 J
   2. B) 5 J
   3. C) 6 J
   4. D) 7 J

7. A variable force $F(x) = 6x - 18$ acts on a particle. Calculate the work done by this force when the particle moves from $x = 3$ m to $x = 6$ m.

   1. A) 27 J
   2. B) 36 J
   3. C) 45 J
   4. D) 54 J