# AP Physics C: Mechanics - Energy & Potential Review 🚀 Hey there, future physicist! Let's get you prepped for the exam with a high-impact review of energy and potential. We'll break down the key concepts, connect them, and make sure you're feeling confident. Let's dive in! ## Hooke's Law and Springs ### The Basics - **Hooke's Law** describes the force exerted by an ideal spring: $$F_s = -k \Delta x$$ - $F_s$ is the spring force (a **restoring force**) - $k$ is the **spring constant** (stiffness of the spring) - $\Delta x$ is the displacement from equilibrium Remember: A larger spring constant ($k$) means a stiffer spring! Think of it like this: a really stiff spring is hard to stretch, so it needs a big force for even a small displacement. - **Spring Constant (k):** A measure of a spring's stiffness. A smaller *k* means the spring is easier to stretch. ### Visualizing Hooke's Law - **Graphing:** When you graph spring force ($F_s$) vs. displacement ($\Delta x$), the slope is the spring constant (*k*). ![Force vs Displacement Graph](https://zupay.blob.core.windows.net/resources/files/0baca4f69800419293b4c75aa2870acd_781fe1_3200.png?alt=media&token=241eaddd-1a2d-453c-928f-b50fe02198d2) *Caption: The slope of this graph represents the spring constant, k.* ### Elastic Potential Energy - The energy stored in a spring due to its deformation: $$U_s = \frac{1}{2}k(\Delta x)^2$$ This formula is crucial! It connects spring displacement to stored energy. ![Elastic Potential Energy](https://zupay.blob.core.windows.net/resources/files/0baca4f69800419293b4c75aa2870acd_ce75bc_2072.png?alt=media&token=f2cb4d2c-4a70-43f2-bb78-d19230197477) ## Conservative Forces ### What are they? - **Conservative Force:** Work done by the force is independent of the path taken. It only depends on the initial and final positions. 💡 - **Dissipative Forces:** Forces like friction or external applied forces where energy is lost from the system (usually as heat). ### Key Characteristics - **Path Independence:** Work done is the same regardless of the path. - **Zero Work in Closed Path:** Total work done around a closed path is zero. ### Examples - **Gravitational Force** - **Spring Force** ### Work and Potential Energy - Work done by a conservative force is equal to the negative change in potential energy: $$W = -\Delta U$$ ![Work Done by Conservative Force](https://zupay.blob.core.windows.net/resources/files/0baca4f69800419293b4c75aa2870acd_736707_1755.png?alt=media&token=42627851-9ff5-4a38-918c-5447ccd1e2dd) - Or, in integral form: $$W = \int_{a}^{b} F \cdot dr = -\Delta U$$ ![Work Integral Form](https://zupay.blob.core.windows.net/resources/files/0baca4f69800419293b4c75aa2870acd_2eb1da_2771.png?alt=media&token=29338695-467d-48a9-a17c-e2cbcce1f187) - Differential version (force as the negative gradient of potential energy): $$F = -\frac{dU}{dr}$$ ![Differential Version](https://zupay.blob.core.windows.net/resources/files/0baca4f69800419293b4c75aa2870acd_eb6df2_621.png?alt=media&token=0a06e636-dca9-4d88-a3e3-492b9725ede1) ### Analyzing Potential Energy Graphs - **Force is the negative slope** of a potential energy vs. position graph. ![Potential Energy Graph](https://zupay.blob.core.windows.net/resources/files/0baca4f69800419293b4c75aa2870acd_61ac86_693.png?alt=media&token=f5508fe8-07b9-47c3-8cbb-7d7a23622e1d) *Caption: The slope of the potential energy graph gives you the negative of the force.* - **Equilibrium:** Occurs where the slope is zero (no net force). - **Total Mechanical Energy:** Draw a horizontal line on the graph to represent it. - **Potential Energy Wells:** Local minimums on the graph, often associated with oscillations. ## Gravitational Potential Energy ### Near Earth (Uniform Field) - Change in gravitational potential energy: $$\Delta U = mg\Delta h$$ - *m* is mass - *g* is acceleration due to gravity - $\Delta h$ is the change in height ![Gravitational Potential Energy](https://zupay.blob.core.windows.net/resources/files/0baca4f69800419293b4c75aa2870acd_ae3226_4278.png?alt=media&token=f694a3df-e86c-4cad-924c-2bba293ddbed) ### General Case (Large Distances) - Derivation: $$W = \int_{r_i}^{r_f} F(r) dr$$ ![Work Integral](https://zupay.blob.core.windows.net/resources/files/0baca4f69800419293b4c75aa2870acd_27b4a8_4273.png?alt=media&token=76c5f8fe-0069-4b51-b1e5-23856e3bd317) - Using Newton's Law of Universal Gravitation: $$F(r) = \frac{Gm_1m_2}{r^2}$$ ![Newton's Law](https://zupay.blob.core.windows.net/resources/files/0baca4f69800419293b4c75aa2870acd_6f00f3_2014.png?alt=media&token=d01f2f36-1633-47b1-883b-b80eca1510b6) - Integrating: $$\Delta U = \int_{r_i}^{r_f} \frac{Gm_1m_2}{r^2} dr$$ ![Integrating](https://zupay.blob.core.windows.net/resources/files/0baca4f69800419293b4c75aa2870acd_1cda3c_2733.png?alt=media&token=202a131c-73e2-4b9a-95e4-303d94fc2aec) - Since gravity is a conservative force: $$W = -\Delta U$$ ![Conservative Force](https://zupay.blob.core.windows.net/resources/files/0baca4f69800419293b4c75aa2870acd_c73278_2816.png?alt=media&token=eeb847fe-ca72-4057-a9ac-69c2c621746f) - Setting initial potential energy to 0 at infinity: $$U = -\frac{Gm_1m_2}{r}$$ ![Final Formula](https://zupay.blob.core.windows.net/resources/files/0baca4f69800419293b4c75aa2870acd_8b2bd1_244.png?alt=media&token=cb63b698-6e33-4ffe-8446-a3ef94bd1700) This is the formula you'll find on your formula sheet! ## Final Exam Focus ### Key Topics - **Hooke's Law & Spring Energy:** Be ready to use these in energy conservation problems. - **Conservative Forces:** Understand the path-independence and relationship to potential energy. - **Potential Energy Graphs:** Master interpreting slopes, equilibrium, and energy wells. - **Gravitational Potential Energy:** Know both the near-Earth and general formulas. ### Common Question Types - **Energy Conservation:** Problems involving springs, gravity, and other conservative forces. - **Graphical Analysis:** Interpreting potential energy graphs to find force and equilibrium points. - **Derivations:** Be familiar with the derivation of potential energy formulas. ### Last-Minute Tips - **Time Management:** Quickly identify the core concepts in each problem. - **Units:** Always double-check your units! - **Free Response:** Show all your work, even if you're not sure of the final answer. Partial credit is your friend! - **Multiple Choice:** Use process of elimination and estimation when needed. ## Practice Questions ### Multiple Choice Questions 1. A spring with a spring constant *k* is compressed by a distance *x*. What is the potential energy stored in the spring? (A) *kx* (B) 1/2 *kx* (C) *kx*² (D) 1/2 *kx*² 2. A ball is thrown vertically upwards. Which of the following is true about the gravitational potential energy of the ball as it goes up? (A) It decreases (B) It increases (C) It remains constant (D) It first increases then decreases 3. Which of the following forces is a non-conservative force? (A) Gravitational force (B) Spring force (C) Frictional force (D) Electric force ### Free Response Question A block of mass *m* is placed against a spring with a spring constant *k*, compressing it a distance *x*. The block is then released, and it slides along a frictionless horizontal surface and then up a frictionless ramp that makes an angle θ with the horizontal. The block eventually reaches a maximum height *h* on the ramp. (a) Determine the potential energy stored in the spring before the block is released. (b) Determine the speed of the block just as it leaves the spring. (c) Determine the maximum height *h* that the block reaches on the ramp. (d) If the surface and ramp were not frictionless, how would this affect the maximum height *h*? Explain your answer. ### Answers #### Multiple Choice Answers 1. (D) 1/2 *kx*² 2. (B) It increases 3. (C) Frictional force #### Free Response Question Scoring (a) 1 point * Correct formula for elastic potential energy: $U = \frac{1}{2}kx^2$ (b) 2 points * Setting the potential energy of the spring equal to the kinetic energy of the block: $\frac{1}{2}kx^2 = \frac{1}{2}mv^2$ * Solving for *v*: $v = \sqrt{\frac{kx^2}{m}}$ (c) 2 points * Setting the kinetic energy of the block equal to the gravitational potential energy: $\frac{1}{2}mv^2 = mgh$ * Solving for *h*: $h = \frac{v^2}{2g} = \frac{kx^2}{2mg}$ (d) 2 points * The maximum height *h* would be less. * Because some of the mechanical energy would be converted to thermal energy due to friction, so less energy would be available to increase the height. ### Multiple Choice Questions 1. A 5.00 × 10^5-kg subway train is brought to a stop from a speed of 0.500 m/s in 0.400 m by a large spring bumper at the end of its track. What is the force constant k of the spring? (A) 1.56 x 10^5 N/m (B) 1.56 x 10^6 N/m (C) 7.81 x 10^5 N/m (D) 7.81 x 10^6 N/m 2. Suppose a 350-g kookaburra (a large kingfisher bird) picks up a 75-g snake and raises it 2.5 m from the ground to a branch. How much work did the bird do on the snake? (A) 1.84 J (B) 18.4 J (C) 184 J (D) 1840 J ### Free Response Question A particle of mass *m* moves along the x-axis. Its potential energy *U(x)* is given by the function *U(x) = ax^3 - bx*, where *a* and *b* are positive constants. (a) Determine the force *F(x)* acting on the particle as a function of position *x*. (b) Determine the equilibrium positions of the particle. (c) Determine whether each equilibrium position is stable or unstable. (d) Sketch the potential energy function *U(x)* and indicate the equilibrium positions on your sketch. ### Answers #### Multiple Choice Answers 1. (A) 1.56 x 10^5 N/m 2. (A) 1.84 J #### Free Response Question Scoring (a) 2 points * Using the relationship between force and potential energy: $F(x) = - \frac{dU}{dx}$ * Correctly taking the derivative: $F(x) = - (3ax^2 - b) = b - 3ax^2$ (b) 2 points * Setting the force equal to zero: $b - 3ax^2 = 0$ * Solving for *x*: $x = \pm \sqrt{\frac{b}{3a}}$ (c) 2 points * Taking the second derivative of the potential energy: $\frac{d^2U}{dx^2} = 6ax$ * Determining the stability: - At $x = \sqrt{\frac{b}{3a}}$, the second derivative is positive, so it is a stable equilibrium. - At $x = -\sqrt{\frac{b}{3a}}$, the second derivative is negative, so it is an unstable equilibrium. (d) 1 point * A sketch of a cubic function with a local minimum at $x = \sqrt{\frac{b}{3a}}$ and a local maximum at $x = -\sqrt{\frac{b}{3a}}$ ### Multiple Choice Questions 1. A spring is compressed by a distance x, and the potential energy stored in it is U. If the spring is compressed by a distance 2x, the potential energy stored in the spring will be: (A) U (B) 2U (C) 4U (D) 8U 2. A satellite of mass m is orbiting the Earth at a distance r from the center of the Earth. If the mass of the Earth is M, what is the gravitational potential energy of the satellite? (A) -GMm/r (B) -GMm/r^2 (C) GMm/r (D) GMm/r^2 ### Free Response Question A small block of mass *m* is released from rest at the top of a curved frictionless ramp of height *h*. At the bottom of the ramp, the block slides onto a horizontal frictionless surface and then collides with a spring with spring constant *k*. (a) Determine the speed of the block at the bottom of the ramp. (b) Determine the maximum compression of the spring due to the collision with the block. (c) If the horizontal surface were not frictionless, how would this affect the maximum compression of the spring? Explain your answer. ### Answers #### Multiple Choice Answers 1. (C) 4U 2. (A) -GMm/r #### Free Response Question Scoring (a) 2 points * Setting the gravitational potential energy at the top of the ramp equal to the kinetic energy at the bottom: $mgh = \frac{1}{2}mv^2$ * Solving for *v*: $v = \sqrt{2gh}$ (b) 2 points * Setting the kinetic energy of the block equal to the elastic potential energy of the spring: $\frac{1}{2}mv^2 = \frac{1}{2}kx^2$ * Solving for *x*: $x = \sqrt{\frac{mv^2}{k}} = \sqrt{\frac{2mgh}{k}}$ (c) 2 points * The maximum compression of the spring would be less. * Because some of the mechanical energy would be converted to thermal energy due to friction, so less energy would be available to compress the spring.

Forces and Potential Energy

Mary Brown

How are we doing?

Forces and Potential Energy

Mary Brown

How are we doing?