# AP Physics C: Mechanics - Unit 3 Study Guide: Work, Energy, and Power 🚀 Hey there, future physics pro! This guide is your go-to resource for acing Unit 3. We'll break down work, energy, and power, making sure you're not just memorizing formulas, but truly understanding the concepts. Let's get started! ## 1. Work-Energy Theorem ### What it is: The **work-energy theorem** is your secret weapon for connecting work and kinetic energy. It states that the net work done on an object equals the change in its kinetic energy. Think of it as the energy transfer in motion. 💡 * **Formula:** $W_{net} = \Delta KE = KE_f - KE_i$ where $KE = \frac{1}{2}mv^2$ * $W_{net}$ is the net work done * $\Delta KE$ is the change in kinetic energy * $KE_f$ is the final kinetic energy * $KE_i$ is the initial kinetic energy The work-energy theorem is a scalar equation, meaning it doesn't involve direction. It's all about the magnitude of energy transfer. ### Key Ideas: * **Work:** Work is done when a force causes a displacement. If the force and displacement are in the same direction, the work is positive. If they're opposite, the work is negative. If they are perpendicular, work done is zero. * **Formula:** $W = F \cdot d \cdot cos(\theta)$ where $\theta$ is the angle between the force and the displacement. * **Kinetic Energy (KE):** The energy of motion. The faster an object moves, the more KE it has. Remember: Work is like the 'push' that changes an object's motion (kinetic energy). ### Example: Imagine pushing a box across a floor. The work you do increases the box's kinetic energy, making it move faster. If you push in the opposite direction of motion, you are doing negative work, slowing it down. **Multiple Choice Questions:** 1. A 2 kg block is pushed along a horizontal surface with a force of 10 N over a distance of 5 m. If the coefficient of kinetic friction is 0.2, what is the net work done on the block? (A) 50 J (B) 40.4 J (C) 10 J (D) 9.6 J 2. A 1 kg ball is dropped from a height of 10 m. What is the kinetic energy of the ball just before it hits the ground? (Assume no air resistance) (A) 98 J (B) 49 J (C) 196 J (D) 0 J **Free Response Question:** A 0.5 kg block is initially at rest on a horizontal surface. A force of 20 N is applied at an angle of 30 degrees above the horizontal. The block moves 2 meters along the surface. The coefficient of kinetic friction between the block and the surface is 0.1. (a) Draw a free body diagram for the block. (b) Calculate the work done by the applied force. (c) Calculate the work done by the frictional force. (d) Calculate the net work done on the block. (e) Calculate the final speed of the block using the work-energy theorem. **Answer Key:** **Multiple Choice:** 1. (B) 40.4 J. Net work = Work by applied force - Work by friction. Work by applied force = 10 N * 5 m = 50 J. Frictional force = μ * normal force = 0.2 * 2 kg * 9.8 m/s^2 = 3.92 N. Work by friction = 3.92 N * 5 m = 19.6 J. Net work = 50 J - 19.6 J = 40.4 J 2. (A) 98 J. The potential energy at the top is converted to kinetic energy at the bottom. Potential energy = mgh = 1 kg * 9.8 m/s^2 * 10 m = 98 J. **Free Response:** (a) Free body diagram should include: gravity (downward), normal force (upward), applied force (at 30 degrees), and friction (opposite to motion). (b) Work by applied force = F * d * cos(θ) = 20 N * 2 m * cos(30) = 34.64 J [2 points] (c) Frictional force = μ * normal force. Normal force = mg - Fsin(θ) = 0.5 * 9.8 - 20 * sin(30) = 4.9 - 10 = -5.1 N. Frictional force = 0.1 * 5.1 = 0.51 N. Work by friction = -0.51 N * 2 m = -1.02 J [3 points] (d) Net work = Work by applied force + Work by friction = 34.64 J - 1.02 J = 33.62 J [2 points] (e) Using the work-energy theorem: Net work = ΔKE. 33.62 J = 1/2 * 0.5 * v^2. v = 11.6 m/s [3 points] ## 2. Forces and Potential Energy ### What it is: **Potential energy** is stored energy due to an object's position or configuration. Forces can do work to change an object's potential energy. When a force does work, it can either increase or decrease the potential energy of a system. 🏞️ * **Gravitational Potential Energy (U_g):** Energy due to height above a reference point. Formula: $U_g = mgh$ * $m$ is mass * $g$ is acceleration due to gravity * $h$ is height * **Elastic Potential Energy (U_s):** Energy stored in a spring when it's stretched or compressed. Formula: $U_s = \frac{1}{2}kx^2$ * $k$ is the spring constant * $x$ is the displacement from equilibrium ### Key Ideas: * **Conservative Forces:** Forces like gravity and spring forces where the work done is independent of the path taken. The work done by a conservative force is equal to the negative change in potential energy. $W = -\Delta U$ * **Non-Conservative Forces:** Forces like friction where the work done depends on the path taken. These forces dissipate energy as heat or other forms. Remember that the work done by a conservative force is equal to the negative change in potential energy. This is a crucial concept for problem-solving. ### Example: Lifting a book increases its gravitational potential energy. Compressing a spring stores elastic potential energy. When you release the book or spring, this potential energy is converted into kinetic energy. **Multiple Choice Questions:** 1. A spring with a spring constant of 200 N/m is compressed by 0.1 m. What is the elastic potential energy stored in the spring? (A) 1 J (B) 2 J (C) 10 J (D) 20 J 2. A 3 kg ball is lifted from the ground to a height of 5 m. What is the change in the gravitational potential energy of the ball? (A) 147.15 J (B) 15 J (C) 14.715 J (D) 150 J **Free Response Question:** A 2 kg block is placed on a frictionless ramp inclined at 30 degrees to the horizontal. The block is attached to a spring with a spring constant of 100 N/m. The block is released from rest when the spring is unstretched. (a) Draw a free body diagram for the block. (b) Determine the distance the block moves down the ramp before momentarily coming to rest. (c) Determine the maximum speed of the block as it moves down the ramp. **Answer Key:** **Multiple Choice:** 1. (A) 1 J. Elastic potential energy = 1/2 * k * x^2 = 1/2 * 200 N/m * (0.1 m)^2 = 1 J 2. (A) 147.15 J. Change in gravitational potential energy = mgh = 3 kg * 9.81 m/s^2 * 5 m = 147.15 J **Free Response:** (a) Free body diagram should include: gravity (downward), normal force (perpendicular to the ramp), and spring force (up the ramp). (b) At the point where the block momentarily comes to rest, the change in gravitational potential energy equals the elastic potential energy stored in the spring. mgh = 1/2kx^2. h=xsin(30). 2 * 9.8 * x * sin(30) = 1/2 * 100 * x^2. x = 0.392 m [5 points] (c) The maximum speed occurs when the net force on the block is zero. At this point, the change in potential energy is converted to kinetic energy. 1/2 * k * x^2 = 1/2 * m * v^2. v = 1.4 m/s [5 points] ## 3. Conservation of Energy ### What it is: **Conservation of energy** is a fundamental principle that states that the total energy of an isolated system remains constant. Energy can transform from one form to another, but it cannot be created or destroyed. This is a powerful tool for analyzing motion without worrying about the details of forces. 🔄 * **Formula:** $KE_i + U_i = KE_f + U_f + W_{nc}$ * $KE_i$ is the initial kinetic energy * $U_i$ is the initial potential energy * $KE_f$ is the final kinetic energy * $U_f$ is the final potential energy * $W_{nc}$ is the work done by non-conservative forces (like friction) Conservation of energy is a cornerstone of physics. It's essential for solving problems involving motion, especially when non-conservative forces are absent or negligible. ### Key Ideas: * **Closed System:** A system where no energy is transferred into or out of the system. * **Energy Transformation:** Energy can change forms (e.g., potential to kinetic, kinetic to thermal), but the total amount remains the same. Don't forget to account for non-conservative forces! If friction or other forces are present, they will reduce the total mechanical energy of the system. ### Example: A roller coaster converts potential energy at the top of a hill into kinetic energy at the bottom. Some energy is lost to friction and air resistance, but the total energy of the system remains constant. **Multiple Choice Questions:** 1. A 1 kg ball is released from rest at a height of 10 m on a frictionless track. What is the speed of the ball when it reaches a height of 5 m? (A) 9.9 m/s (B) 7.0 m/s (C) 14 m/s (D) 10 m/s 2. A pendulum is released from an angle of 30 degrees with the vertical. At the bottom of its swing, what form of energy does it have? (A) Potential energy (B) Kinetic energy (C) Both potential and kinetic energy (D) Thermal energy **Free Response Question:** A 0.2 kg block slides down a curved, frictionless track from a height of 3 meters. At the bottom of the track, it encounters a rough horizontal surface with a coefficient of kinetic friction of 0.3. (a) Calculate the speed of the block at the bottom of the track. (b) Determine the distance the block slides on the rough surface before coming to rest. (c) How much thermal energy is generated due to friction? **Answer Key:** **Multiple Choice:** 1. (A) 9.9 m/s. Using conservation of energy: mgh_initial = mgh_final + 1/2mv^2. 1 * 9.8 * 10 = 1 * 9.8 * 5 + 1/2 * 1 * v^2. v = 9.9 m/s 2. (B) Kinetic energy. At the bottom of the swing, all potential energy is converted to kinetic energy. **Free Response:** (a) Using conservation of energy: mgh = 1/2mv^2. 0.2 * 9.8 * 3 = 1/2 * 0.2 * v^2. v = 7.67 m/s [3 points] (b) The kinetic energy at the bottom of the track is converted to work done by friction. 1/2mv^2 = f * d. f = μ * normal force = μ * mg. 1/2 * 0.2 * (7.67)^2 = 0.3 * 0.2 * 9.8 * d. d = 10 m [4 points] (c) The thermal energy generated is equal to the work done by friction, which is also equal to the initial kinetic energy of the block at the bottom of the track. Thermal energy = 1/2 * 0.2 * (7.67)^2 = 5.9 J [3 points] ## 4. Power ### What it is: **Power** is the rate at which work is done, or the rate at which energy is transferred. It tells you how quickly work is being done or energy is being used. ⚡ * **Formula:** $P = \frac{W}{t} = \frac{\Delta E}{t}$ * $P$ is power * $W$ is work done * $\Delta E$ is the change in energy * $t$ is time * **Alternative Formula:** $P = F \cdot v$ * $F$ is the force * $v$ is the velocity Power is measured in watts (W), where 1 watt = 1 joule/second. ### Key Ideas: * **Average Power:** The total work done divided by the total time. * **Instantaneous Power:** The power at a specific moment in time. ### Example: A powerful car engine can do more work (accelerate faster) in the same amount of time compared to a less powerful engine. A lightbulb with higher power rating consumes more electrical energy per second. **Multiple Choice Questions:** 1. A machine does 1000 J of work in 5 seconds. What is the power output of the machine? (A) 200 W (B) 5000 W (C) 2000 W (D) 500 W 2. A car is moving at a constant speed of 20 m/s and the engine provides a force of 5000 N. What is the power output of the engine? (A) 100 kW (B) 250 kW (C) 10 kW (D) 25 kW **Free Response Question:** A 1000 kg elevator is lifted vertically at a constant speed of 2 m/s. (a) What is the power output of the motor lifting the elevator? (b) If the elevator is lifted to a height of 20 meters, how much work is done by the motor? (c) How much time does it take to lift the elevator to a height of 20 meters? **Answer Key:** **Multiple Choice:** 1. (A) 200 W. Power = Work / Time = 1000 J / 5 s = 200 W 2. (A) 100 kW. Power = Force * Velocity = 5000 N * 20 m/s = 100000 W = 100 kW **Free Response:** (a) Power = Force * Velocity. Force = mg = 1000 kg * 9.8 m/s^2 = 9800 N. Power = 9800 N * 2 m/s = 19600 W [3 points] (b) Work = Force * Distance = 9800 N * 20 m = 196000 J [3 points] (c) Time = Distance / Velocity = 20 m / 2 m/s = 10 s [4 points] ## Final Exam Focus 🎯 Alright, let's talk about what you absolutely need to nail on the exam: * **Work-Energy Theorem:** Understand how work changes kinetic energy. Practice problems involving variable forces. * **Potential Energy:** Know how to calculate gravitational and elastic potential energy. Be comfortable with conservative and non-conservative forces. * **Conservation of Energy:** This is HUGE. Apply it to various scenarios, including those with and without friction. Look out for problems combining potential and kinetic energy. * **Power:** Be ready to calculate power using work/time and force/velocity. Understand the difference between average and instantaneous power. ### Last-Minute Tips: * **Time Management:** Don't spend too long on a single question. Move on and come back if needed. Prioritize questions you know you can solve quickly. * **Units:** Always include units in your answers. Pay attention to conversions (e.g., cm to m, grams to kg). * **Free Body Diagrams:** Draw them! They help visualize forces and are often worth points on FRQs. * **Practice, Practice, Practice:** The more problems you solve, the more confident you'll feel. Review your mistakes and understand why you made them. You've got this! Remember, physics is about understanding the world around you. Approach the exam with confidence and a clear mind. Good luck! 🎉

Work, Energy, and Power

Robert Jones

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Work, Energy, and Power

Robert Jones

How are we doing?