Work and Kinetic Energy

Introduction

This chapter introduces the concepts of work and kinetic energy, fundamental to understanding how forces transfer energy to objects. The scientific definition of work connects force, displacement, and energy transfer, providing a basis for analyzing mechanical systems.

Work

Definition of Work

• Work is done on a system when an applied force results in a net displacement of the system.

• If a force is applied but there is no displacement, no work is done.

• If there is displacement without an applied force (e.g., orbital motion), no work is done by that force.

Scientific definition: Whenever work is done, energy is transferred.

Work Done by a Constant Force

• For an object moving in one dimension under a constant force, the work done is:

• = magnitude of the constant force

• = magnitude of the displacement in the direction of the force

When the force is not aligned with the displacement, only the component of the force in the direction of displacement does work:

• = angle between the force and displacement vectors

This can be written as the dot product:

Units of Work

• 1 J = 0.7376 ft·lb

Work: Positive, Negative, or Zero

• Work is a scalar quantity (it has magnitude but no direction).

• Positive work: Force and displacement are in the same direction ().

• Negative work: Force and displacement are in opposite directions ().

• Zero work: Force is perpendicular to displacement (), or displacement is zero.

Examples

• Lifting a box: Work done by the person is positive (force and displacement upward).

• Lowering a box: Work done by the person is negative (force upward, displacement downward).

• Holding a barbell stationary: No work is done (displacement is zero).

• Work done by gravity: Positive when an object is dropped (force and displacement downward), negative when lifted upward.

Work Done by a Constant Force at an Angle

• When a force acts at an angle to the displacement:

• Only the component of the force in the direction of displacement does work.

Example: Pulling a Box

• A 60 N force is applied at a 35° angle to pull a box 45 m on a level floor:

Solving Work Problems

1. Draw a free-body diagram.

2. Choose a coordinate system.

3. Apply Newton's laws to determine unknown forces.

4. Find the work done by each force.

5. To find net work, either:

   • Find the net force and calculate the work it does, or

   • Find the work done by each force and sum them.

Example: Net Work on a Package

• Push a 30.0-kg package with 120 N over 0.800 m, friction force 5.00 N.

• Net force:

• Net work:

Practice Problems

• Child pulls a wagon: 16.0 N for 10.0 m

  • Parallel to floor:

  • At 25.0°:

• Baseball catcher: Work done by the catcher is negative, as the force is opposite to the ball's displacement.

Work Done by Multiple Forces

Inclined Plane and Multiple Forces

• When an object is pulled up an incline, several forces may do work: tension, friction, gravity, and the normal force.

• Only forces with a component along the displacement do work.

For each force acting at angle to the displacement :

Example: Pulling a Crate

• 50-kg crate, N at 37°, friction N, m.

• Work by pulling force:

• Work by friction:

• Work by gravity and normal force: $0$ (perpendicular to displacement)

• Net work:

Kinetic Energy

Definition of Kinetic Energy

• Kinetic energy (KE) is the energy associated with the motion of an object.

• It is a scalar quantity and has the same units as work (joules).

The kinetic energy of an object of mass moving at speed is:

Example

• A 7.00 kg bowling ball moves at 3.00 m/s. Its kinetic energy is:

Comparing Kinetic Energies

• For two objects with different masses and speeds, the one with greater has more kinetic energy.

• If ball A has half the mass and eight times the kinetic energy of ball B, the speed ratio can be found by solving:

with

Summary Table: Work by Force Direction

Key Takeaways

• Work is the transfer of energy via force and displacement.

• Work can be positive, negative, or zero depending on the angle between force and displacement.

• Kinetic energy quantifies the energy of motion and is directly related to work done on an object.

• Solving work problems requires careful identification of forces, directions, and the use of vector components.