Work-Energy Theorem

Unit 5: Energy and Work

Work, PE and KE

Work-Energy Theorem

• In order to change an object’s motion, you must do work on the object.
• Reminder: You do work on an object by applying a force over a distance
• The force must be applied parallel to the object’s motion

Work-Energy Theorem

• States: The net work done by a net force acting on an object is equal to the change in the kinetic energy of the object

W = ΔKE

Sample Problem 1

A student wearing frictionless in-line skates on a horizontal surface is pushed by a friend with a constant 45 N force. How far must the student be pushed, starting from rest, so that her final KE is 352 J?

F = 45N

KE_i = 0 J

KE_f = 352 J

d = ?

W = ΔKE

Fd = KE_f – KE_i

45d = 352 – 0

d = 7.82 m

Sample Problem 2

A 75 kg bobsled is pushed along a horizontal surface by two athletes. After the bobsled is pushed a distance of 4.5 m starting from rest, its final speed is 6 m/s. Find the net force on the bobsled.

m = 75 kg

F = ?

KE_i = 0 J

d = 4.5 m

v_f = 6 m/s

W = ΔKE

Fd = KE_f – KE_i

Fd = ½ mv_f² – KE_i

4.5F = ½(75)(6²) – 0

4.5F = 1350

F = 300 N

Sample Problem 3

A 2.5 kg book is moving at 7.3 m/s initially. The book stops in 3 meters. Calculate the frictional force on the book.

m = 2.5 kg

F = ?

KE_f = 0 J

d = 3 m

v_i = 7.3 m/s

W = ΔKE

Fd = KE_f – KE_i

Fd = KE_f – ½ mv_i²

3F = 0 – ½(2.5)(7.3²)

3F = -66.6

F_f = -22.2 N

Why is it negative??

Consider the Situation…

You are on a roller coaster that starts 40 meters high in a cart that has a total mass of 250 kg.

What is the TME at the top of the hill?

TME = KE + PE 🡪 TME = 0 + mgh 🡪 TME = (250)(9.8)(40) 🡪 TME = 98,000 J

Eventually the roller coaster stops on level ground. According to the Law of Conservation of energy, this should not happen. Your TME at the bottom should be equal to the TME at the top.

Why aren’t they equal?

Consider the Situation…

When work is applied, the system is no longer closed. By friction acting on the roller coaster causing it to stop, the energy is no longer conserved within the system.

Work-Energy Bar Charts

• Take this same roller coaster situation. A roller coaster is at the top of a hill before going down. Friction is applied to stop the roller coaster at the bottom of the hill.
• How do we represent this situation as an energy-bar chart INCLUDING work?

Final energy = 0 J because it is on the ground at rest.

Work-Energy Bar Charts

• A roller skate is rolling across the floor at a certain velocity and you push it to increase its velocity.
• How do we represent this as a work-energy bar chart?

Final energy = KE + W

Lesson Check 5.4