Unit 3: Energy
Work, Kinetic & Potential Energy, Conservation, and Power
Kinetic Energy & Work
Work and Translational Kinetic Energy
Work is the transfer of energy by a force: \( W = \int \vec{F} \cdot d\vec{r} \). The Work-Energy Theorem states that the net work done equals the change in kinetic energy: \( W_{net} = \Delta K \).
Q1: If the speed of an object is tripled, by what factor does its kinetic energy increase?
Kinetic energy is defined as \( K = \frac{1}{2}mv^2 \). Since \( K \propto v^2 \), if velocity increases by a factor of 3, kinetic energy increases by \( 3^2 = 9 \).
Correct Answer: C
Correct Answer: C
Q2: A force \( F \) at an angle \( \theta \) above the horizontal pulls a block a distance \( d \). What is the work done?
Work is the dot product of force and displacement: \( W = \vec{F} \cdot \vec{d} = Fd \cos\theta \). Only the component of force parallel to displacement does work.
Correct Answer: C
Correct Answer: C
Q3: How much work is done by a force \( F(x) = ax^2 \) moving a particle from \( x = 0 \) to \( x = L \)?
For a variable force: \( W = \int_0^L F(x) dx = \int_0^L ax^2 dx \).
Integrating yields: \( [\displaystyle\frac{1}{3}ax^3]_0^L = \displaystyle\frac{1}{3}aL^3 \).
Correct Answer: D
Integrating yields: \( [\displaystyle\frac{1}{3}ax^3]_0^L = \displaystyle\frac{1}{3}aL^3 \).
Correct Answer: D
Potential Energy & Conservation
Energy Conservation
Conservative forces like gravity and springs allow us to define Potential Energy (\( U \)). The relationship is \( F = -\displaystyle\frac{dU}{dx} \). Mechanical energy \( E = K + U \) is conserved when only conservative forces do work.
Q1: Given \( U(x) = 4x^2 – 2x \), what is the force acting on the particle at \( x = 1 \)?
Force is the negative gradient of potential energy: \( F = -\displaystyle\frac{dU}{dx} \).
\( \displaystyle\frac{dU}{dx} = 8x – 2 \).
\( F = -(8x – 2) = 2 – 8x \). At \( x = 1 \): \( F = 2 – 8 = -6 \text{ N} \).
Correct Answer: B
\( \displaystyle\frac{dU}{dx} = 8x – 2 \).
\( F = -(8x – 2) = 2 – 8x \). At \( x = 1 \): \( F = 2 – 8 = -6 \text{ N} \).
Correct Answer: B
Q2: What is the minimum height \( H \) for a block starting from rest to complete a loop-the-loop of radius \( R \)?
At the top of the loop, the minimum speed is \( v = \sqrt{gR} \).
Energy at the start: \( mgH \).
Energy at top of loop: \( mg(2R) + \displaystyle\frac{1}{2}m(\sqrt{gR})^2 = 2mgR + 0.5mgR = 2.5mgR \).
Conservation gives: \( mgH = 2.5mgR \Rightarrow H = 2.5R \).
Correct Answer: B
Energy at the start: \( mgH \).
Energy at top of loop: \( mg(2R) + \displaystyle\frac{1}{2}m(\sqrt{gR})^2 = 2mgR + 0.5mgR = 2.5mgR \).
Conservation gives: \( mgH = 2.5mgR \Rightarrow H = 2.5R \).
Correct Answer: B
Power
Power
Power is the rate of doing work: \( P = \displaystyle\frac{dW}{dt} \). For a force acting on an object with velocity \( v \), instantaneous power is \( P = \vec{F} \cdot \vec{v} \).
Q1: A motor lifts 100 kg at a constant speed of 2 m/s. What is the power output? (Use \( g = 10 \text{ m/s}^2 \))
At constant speed, \( F_{motor} = mg = 100 \times 10 = 1000 \text{ N} \).
\( P = Fv = 1000 \times 2 = 2000 \text{ W} \).
Correct Answer: C
\( P = Fv = 1000 \times 2 = 2000 \text{ W} \).
Correct Answer: C
Q2: If work is given by \( W(t) = 5t^2 + 2t \), what is the instantaneous power at \( t = 3 \text{ s} \)?
\( P = \displaystyle\frac{dW}{dt} = \displaystyle\frac{d}{dt}(5t^2 + 2t) = 10t + 2 \).
At \( t = 3 \): \( P = 10(3) + 2 = 32 \text{ W} \).
Correct Answer: B
At \( t = 3 \): \( P = 10(3) + 2 = 32 \text{ W} \).
Correct Answer: B
Energy Formulas Recap
Kinetic Energy\( K = \displaystyle\frac{1}{2}mv^2 \)
Work (General)\( W = \int \vec{F} \cdot d\vec{r} \)
Potential Energy\( U_g = mgh \) | \( U_s = \frac{1}{2}kx^2 \)
Force/Potential\( F = -\displaystyle\frac{dU}{dx} \)
Power (Force)\( P = \vec{F} \cdot \vec{v} \)
Power (Work)\( P = \displaystyle\frac{dW}{dt} \)
