C. Energy and Momentum · Work, Energy and Simple Harmonic Motion

Work, Energy and Simple Harmonic Motion: Unit Review

C1 C1.2 C2.1 C2.2 C2.3 C2.4 C3.1 C3.2 C3.4 A1.1 A1.5 A1.6 A1.8 A1.10 A1.13 A2.1About 40 minutes

Learning goals

I can solve work and energy problems in one and two dimensions, with and without friction.
I can use Hooke's law, elastic potential energy and simple harmonic motion for springs and pendulums.
I can explain damping, perpetual motion, power, efficiency and energy technologies.

Words to know

work–energy theorem
W_net = ΔEk.

conservation of energy
Energy is transformed or transferred but never created or destroyed.

simple harmonic motion
Motion with a restoring force proportional to displacement: T = 2π√(m/k) or 2π√(L/g).

damping
Loss of mechanical energy that makes an oscillation's amplitude decrease.

Let's review!

In this unit you defined work for forces at any angle and for changing forces, used the work–energy theorem in two dimensions, stored energy as gravitational and elastic potential energy, and applied the law of conservation of energy with and without friction. You studied Hooke's law, simple harmonic motion of springs and pendulums and its link to uniform circular motion, tested conservation of energy by investigation, explained damping and the impossibility of perpetual motion, and used power, efficiency and energy principles to analyse real technologies.

The next unit, Momentum and Collisions, adds a second conservation law. This review has a few questions from every lesson. Try each one, then press Show answer to check. Use g = 9.8 N/kg.

How to use this review

Work through one or two sections at a time. For every energy problem, sketch the start and the end, choose a reference level, and list Ek, Eg, Ee and any thermal energy or other work before you calculate. Convert cm to m before squaring, and keep your calculator in radian mode for x = A cos(2πt/T) and in degree mode for angles of forces. For each question you miss, write down why (wrong angle? forgot friction? forgot to square?) and go back to that lesson. Finish with the 10-question quiz.

How the ideas in this unit connect. The numbers in brackets are the lessons where each idea is taught.

The energy equation that ties the unit together. Every energy problem is a special case of it.

Work Done by a Constant Force in One and Two Dimensions

A rope pulls a sled 12 m with a tension of 60 N at 20° above the horizontal. How much work does the rope do?

A box slides down a ramp. How much work does the normal force do on it? Why?

F = (10, −5) N and Δd = (3.0, 4.0) m. Find the work.

Work Done by a Varying Force: The Area Under the Graph

A force rises steadily from 0 to 60 N over 0.40 m. Find the work.

A force decreases steadily from 40 N to 20 N over 0.50 m. Find the work.

What does area below the displacement axis on an F–Δd graph mean?

Kinetic Energy and the Work–Energy Theorem in Two Dimensions

A 3.0 kg cart moving at 2.0 m/s has 24 J of net work done on it. Find its new speed.

A puck slides at 5.0 m/s on ice with μ_k = 0.10. How far does it go before stopping?

Why do two balls thrown from the same height at the same speed, at different angles, land with the same speed (no air resistance)?

Gravitational Potential Energy and Conservative Forces

A 50 kg climber rises 12 m. Find ΔEg.

What makes a force conservative?

A lamp hangs 0.50 m below the ceiling. Using the ceiling as the reference level, is its Eg positive, negative or zero?

Conservation of Mechanical Energy: Tracks, Pendulums and Loops

A ball is dropped from 3.2 m. Ignoring air resistance, find its speed at the ground.

A 0.60 m pendulum is released from rest at 25°. Find its speed at the bottom.

A frictionless loop has radius 4.0 m. From what minimum height must a car start from rest to complete it?

Friction and Thermal Energy: Energy Accounting on Slopes and Tracks

A 40 kg sled starts from rest at the top of a 10 m high slope and reaches the bottom at 11 m/s. How much thermal energy was produced?

A 2.0 kg box slides 3.0 m on a level floor with μ_k = 0.30. Find E_th.

On an incline of angle θ, what is the friction force on a sliding block (no other forces into the surface)?

Hooke's Law: Springs and Restoring Forces

A 12 N force stretches a spring 4.0 cm. Find k.

Two of these springs are hooked end to end. What is the effective spring constant?

A spring is stretched to the right of its equilibrium position. Which way does the spring force point?

Elastic Potential Energy

Find the energy stored in a spring (k = 500 N/m) compressed 6.0 cm.

If the compression is tripled, by what factor does Ee change?

A spring (k = 200 N/m) is stretched from 0.10 m to 0.20 m. How much work is done?

Just for fun

Energy is like a very honest accountant: it never disappears from the books. When it seems to, check the column labelled "thermal".

Springs in Energy Problems: Launchers, Drops and Bungee Cords

A spring (k = 100 N/m) compressed 0.10 m launches a 0.50 kg block on a frictionless surface. Find the block's speed.

Where is a bungee jumper moving fastest?

A block of mass m is dropped from height h above a vertical spring. Write the equation for the maximum compression x.

Simple Harmonic Motion of a Mass on a Spring

Find the period of a 0.30 kg mass on a spring with k = 120 N/m.

The mass is made four times as large. What happens to the period?

Where is the acceleration of a mass in SHM zero?

SHM, Energy and the Reference Circle

At what positions are the kinetic and elastic energies of a mass-spring oscillator equal?

The amplitude of an oscillator is doubled. What happens to its total energy?

A mass is released from x = +A. Where is it at t = T/2?

The Simple Pendulum

Find the period of a 0.60 m pendulum.

A pendulum bob is replaced by one with twice the mass. What happens to the period?

A 0.99 m pendulum has a period of 2.00 s. Find g.

Investigation: Testing Conservation of Energy

A ball dropped from 2.0 m bounces to 1.6 m. What percentage of its energy did it keep?

A predicted speed is 10.0 m/s and the measured speed is 9.5 m/s. Find the percent difference.

Name one controlled variable in a pendulum test of energy conservation.

Damped Harmonic Motion, Shock Absorbers and Perpetual Motion

An oscillator's amplitude drops to 80% in one cycle. What fraction of its energy remains?

Which damping returns a system to equilibrium fastest without overshooting?

Explain in one sentence why a perpetual motion machine cannot deliver work forever.

Power and Efficiency in Mechanical Systems

A 600 kg load is lifted 10 m at constant speed in 20 s. Find the useful power.

A vehicle moves at 15 m/s against a 400 N resistive force. Find the power needed.

A motor draws 2.0 kW and delivers 1.5 kW. Find its efficiency.

Energy Technologies: Analyse, Improve and Assess

What does regenerative braking do with a vehicle's kinetic energy?

An elevator car (1000 kg) and counterweight (800 kg) move 10 m, car going up. How much work does the motor do (no friction)?

Give one environmental impact of pumped-storage hydroelectricity.

Check your understanding

Quiz

A 40 N force acts at 60° to an 8.0 m displacement. The work done is:

The area under a force–displacement graph equals:

A 2.0 kg ball falls from rest through 5.0 m with no air resistance. Its kinetic energy at the bottom is:

Which force is non-conservative?

A spring with k = 400 N/m is stretched 0.10 m. The stored energy is:

The period of a mass-spring oscillator does NOT depend on:

The shadow of uniform circular motion on a diameter shows:

To double the period of a pendulum, its length must be:

A shock absorber reduces bouncing by:

10.

A crane lifts a 1000 kg load 20 m in 40 s. Its useful power is:

For parents and teachers

This review covers the whole unit. Use the mistakes to decide what to revisit:

Wrong angle or sign in work: lesson 1 (Work Done by a Constant Force in One and Two Dimensions).
Trouble with graphs, triangles and trapezoids: lesson 2 (Work Done by a Varying Force: The Area Under the Graph).
Using the work of one force instead of the net work: lesson 3 (Kinetic Energy and the Work–Energy Theorem in Two Dimensions).
Path length instead of height, or reference-level confusion: lesson 4 (Gravitational Potential Energy and Conservative Forces).
Pendulum heights, loops and speed on tracks: lesson 5 (Conservation of Mechanical Energy: Tracks, Pendulums and Loops).
Friction on inclines, E_th on the wrong side: lesson 6 (Friction and Thermal Energy: Energy Accounting on Slopes and Tracks).
Spring constants, series and parallel: lesson 7 (Hooke's Law: Springs and Restoring Forces); ½kx² errors (cm not converted, forgetting to square): lesson 8 (Elastic Potential Energy).
Quadratics in spring drops, bungee problems: lesson 9 (Springs in Energy Problems: Launchers, Drops and Bungee Cords).
Period formulas and where v and a are largest: lesson 10 (Simple Harmonic Motion of a Mass on a Spring); x = A cos(2πt/T), energy in SHM: lesson 11 (SHM, Energy and the Reference Circle); pendulum period: lesson 12 (The Simple Pendulum).
Percent difference, variables, conclusions: lesson 13 (Investigation: Testing Conservation of Energy).
Amplitude versus energy in damping, perpetual motion: lesson 14 (Damped Harmonic Motion, Shock Absorbers and Perpetual Motion).
Power, P = Fv, efficiency: lesson 15 (Power and Efficiency in Mechanical Systems); technology analysis and impacts: lesson 16 (Energy Technologies: Analyse, Improve and Assess).

Encourage your teen to explain each answer out loud, starting with "Energy at the start is... energy at the end is...". If they can tell the energy story, the algebra usually follows.