PHY131H1/Unit 6

Unit 6 · Introduction to Physics I

Work, Energy & Power

The first shortcut. Newton's laws work but need the force at every instant; energy methods need only the start and the end. Problems that would take a page of vector algebra collapse to one line of bookkeeping.

By the end you can

• Compute work, including when force and displacement are not aligned.
• Apply the work-energy theorem.
• Distinguish conservative from non-conservative forces.
• Solve problems by conservation of mechanical energy.

Why it matters & where it comes from

Why it exists

Energy is a single number that survives complicated motion. A block sliding down a curved track has a changing force and a changing direction, both hopeless with Newton directly — but its speed at the bottom takes one line.

In the real world

Every energy bill, engine rating and battery capacity is this unit. Rollercoaster designers work almost entirely in energy, since the track shape makes force analysis intractable.

Story & open questions

Energy conservation was only accepted in the 1840s, after Joule showed that mechanical work and heat are interchangeable. Before that, the energy apparently lost to friction had no home, and the principle looked false.

Work

Definition

For a constant force $F$ acting through a displacement $d$ at angle $\theta$ between them,

$W = Fd\cos\theta$

measured in joules. Work is a scalar — it has a sign but no direction.

Rule Only the aligned component works

$\theta = 0$: full work, $W = Fd$.

$\theta = 90$ degrees: no work at all. A force perpendicular to the motion does nothing.

$\theta = 180$ degrees: negative work, $W = -Fd$. Friction and braking do negative work.

Example Forces that do no work

Carrying a box horizontally at constant speed: your upward force is perpendicular to the motion, so the work is zero — however tired you get. Your muscles consume energy holding it, but that is physiology, not mechanics.

The normal force on a sliding block: always perpendicular to the surface, so it never does work.

Centripetal force in uniform circular motion: perpendicular to velocity, so no work is done and the speed cannot change. That is consistent with the speed being constant.

Kinetic energy and the work-energy theorem

Definition

Kinetic energy is $K = \tfrac12 mv^2$: a scalar, never negative, and depending on speed only, not direction.

Theorem Work-energy theorem

The net work done on a body equals its change in kinetic energy:

$W_{\text{net}} = \Delta K = \tfrac12 mv_f^2 - \tfrac12 mv_i^2$

This holds for any force, constant or not, along any path.

Proof From Newton, in one dimension

$W = Fd = (ma)d$.

From Unit 2, $v_f^2 = v_i^2 + 2ad$, so $ad = \tfrac12(v_f^2 - v_i^2)$.

Substituting, $W = \tfrac12 m(v_f^2 - v_i^2) = \Delta K$. The theorem is Newton's second law with the time eliminated. ∎

Example Braking, again

A $1200$ kg car at $25$ m/s brakes to rest in $40$ m. Find the braking force.

$\Delta K = 0 - \tfrac12(1200)(625) = -375{,}000$ J.

$W = -Fd$, so $F = 375{,}000/40 = 9375$ N.

Doubling the speed quadruples $K$ and so quadruples the stopping distance — the same result as Unit 2, arrived at differently.

Potential energy and conservation

Definition

A force is conservative if the work it does depends only on the endpoints, not the path taken — equivalently, if it does zero net work around any closed loop. Gravity and spring forces are conservative; friction and air resistance are not.

Only conservative forces get a potential energy:

Gravitational: $U = mgh$, with $h$ measured from any chosen reference.

Spring: $U = \tfrac12 kx^2$, with $x$ the displacement from the natural length.

Rule Conservation of mechanical energy

If only conservative forces do work,

$K_i + U_i = K_f + U_f$

With friction present, the shortfall is the work friction did:

$K_i + U_i + W_{\text{friction}} = K_f + U_f$, where $W_{\text{friction}}$ is negative.

Example Why the track shape does not matter

A block slides from rest down a frictionless track of height $h$, of any shape.

$mgh = \tfrac12 mv^2$, so $v = \sqrt{2gh}$.

The mass cancels, and the shape never entered. A straight ramp, a curve and a loop all give the same speed at the bottom, provided friction is absent.

Newton's laws would need the force at every point of the curve. This is the argument for energy methods in one example.

Example With friction

The same block, now with $200$ J lost to friction, starting from $mgh = 500$ J.

$K_f = 500 - 200 = 300$ J, so the block arrives slower. The missing $200$ J became heat in the track and the block.

Total energy is still conserved — only mechanical energy is not. That distinction is what took until the 1840s to establish.

Choose a reference height and keep it

$U = mgh$ has no absolute meaning; only differences matter. Pick $h = 0$ somewhere convenient, usually the lowest point, and use it consistently on both sides of the equation.

Power

Definition

Power is the rate of doing work: $P = W/t$, measured in watts. For a force acting on a body moving at speed $v$ in the same direction, $P = Fv$.

Example Climbing stairs

A $70$ kg person climbs $3$ m in $5$ s.

Work against gravity: $mgh = 70 \times 9.8 \times 3 = 2058$ J.

Power: $2058/5 = 412$ W — about half a horsepower, sustainable only briefly.

Example Cruising car

A car at $30$ m/s against $600$ N of drag needs $P = Fv = 18{,}000$ W, about $24$ horsepower.

Drag grows roughly as $v^2$, so power needed grows as $v^3$. Going half again as fast needs more than three times the power, which is why fuel economy collapses at high speed.

Common pitfalls

• Forgetting $\cos\theta$ when force and displacement are not aligned.
• Claiming work is done by a perpendicular force. It is zero.
• Using conservation of mechanical energy when friction is present.
• Changing the height reference partway through a problem.
• Treating kinetic energy as a vector. It is a scalar, and speeds do not cancel the way velocities do.

Practice

Work each fully before opening the answer. Original exercises written for this guide in the style of the course's problem sets.

Practice 1 · Angled push

A $50$ N force at $60$ degrees to the horizontal drags a crate $10$ m along the floor. Find the work.

Answer

$W = 50 \times 10 \times \cos 60 = 250$ J.

Practice 2 · Zero work

How much work does the normal force do on a sliding block?

Answer

Zero — it is perpendicular to the motion throughout.

Practice 3 · Speed at the bottom

A cart starts from rest at the top of a frictionless $8$ m hill. Find its speed at the bottom.

Answer

$v = \sqrt{2(9.8)(8)} = 12.5$ m/s, whatever the shape of the hill.

Practice 4 · Spring

A spring with $k = 400$ N/m is compressed $0.1$ m. How much energy is stored?

Answer

$\tfrac12(400)(0.01) = 2$ J.

Practice 5 · Launch from a spring

That spring launches a $0.2$ kg ball horizontally. Find its speed.

Solution

All $2$ J becomes kinetic: $\tfrac12(0.2)v^2 = 2$, so $v = 4.5$ m/s. Spring potential converting to kinetic is the whole calculation.

Practice 6 · With friction

A $2$ kg block slides $5$ m down a $30$-degree incline with $\mu_k = 0.25$. Find its speed at the bottom, from rest.

Solution

Height dropped is $5\sin 30 = 2.5$ m, so $U$ released is $2(9.8)(2.5) = 49$ J. Friction is $0.25 \times 2 \times 9.8 \times \cos 30 = 4.24$ N over $5$ m, taking $21.2$ J. So $K = 27.8$ J and $v = \sqrt{2(27.8)/2} = 5.3$ m/s.

Practice 7 · Power to climb

A $1000$ kg lift rises $20$ m in $10$ s at constant speed. Find the power.

Answer

$W = mgh = 196{,}000$ J, so $P = 19{,}600$ W. Constant speed means no change in kinetic energy, so all the work goes into potential.

Practice 8 · Which method

Why solve a curved-track problem with energy rather than Newton?

Answer

Because the direction and magnitude of the net force change continuously along the curve, so Newton would need integration. Energy depends only on the endpoints, so the path drops out entirely.

Expanded topic checklist

Course coverage. This unit is part of the complete PHY131H1 topic map. It should cover the following items before you move on:

• Work. $Fd\cos\theta$, its sign, and the perpendicular case.
• Work-energy theorem. Net work equals change in kinetic energy.
• Conservative forces. Path independence and the potential energies that follow.
• Conservation. Mechanical energy with and without friction, and power as a rate.

Textbook/source alignment

The calendar lists "energy, work, power" here. Check the syllabus on Quercus for assigned problems; energy methods reappear throughout the rest of the course.

Core idea map

Work transfers energy; net work changes kinetic energy; conservative forces store it as potential. Conservation then relates any two moments without touching what happened in between.

Study routine. When a problem mentions speeds and heights but not time, reach for energy. When it mentions forces and time, reach for Newton or impulse. That single triage saves more time than any algebra.

Deep practice extensions

Use these after the main practice set. They push on definitions and edge cases rather than routine calculation.

Extension 1 · Perpendicular force

How much work does a force at $90$ degrees to the motion do?

Guide

None. $\cos 90 = 0$, so the work is zero regardless of the force's size.

Extension 2 · Conservative test

What makes a force conservative?

Guide

Its work depends only on the endpoints, so it does zero work around any closed path.

Extension 3 · Missing energy

Where does energy go when friction acts?

Guide

Into heat. Total energy is conserved; mechanical energy is not.

Extension 4 · Reference height

Does the choice of $h = 0$ affect the answer?

Guide

No, provided it is used consistently. Only differences in $U$ appear in the final result.

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