Work and Kinetic Energy

Learning Outcomes

This chapter introduces the fundamental concepts of work, kinetic energy, and power in physics. Students will learn how forces do work on objects, how to calculate work, the definition and properties of kinetic energy, and how to apply the work-energy theorem to various physical situations.

• Understand what it means for a force to do work on an object and how to calculate the amount of work done.

• Define kinetic energy and relate it to the motion of an object.

• Apply the work-energy theorem to relate work done to changes in kinetic energy.

• Analyze situations involving non-constant forces and curved paths.

• Solve problems involving power, the rate at which work is done.

Introduction to Work, Energy, and Conservation

In many physical situations, such as a baseball pitcher throwing a ball, forces do work to change the energy of objects. The concept of kinetic energy (energy of motion) is central to understanding how forces affect motion. The introduction of work, energy, and the conservation of energy provides powerful tools for solving problems that may be difficult to address using Newton's laws alone.

Work

Definition of Work

A force on an object does work if the object undergoes a displacement. Work is a measure of energy transfer that occurs when an object is moved by a force.

• Mathematical Definition: If a constant force acts on a particle causing a displacement in the same direction, the work done is:

• Work is only done when the force has a component along the direction of displacement.

• Example: Pushing a car through snow involves doing work because the force applied causes the car to move.

Units of Work

The SI unit of work is the joule (J), named after James Prescott Joule.

• 1 joule = 1 newton × 1 meter =

• If you lift an object with a weight of 1 N a distance of 1 m at constant speed, you do 1 J of work.

Work Done by a Constant Force

When a constant force acts at an angle to the displacement, the work done is:

• is the angle between the force and the displacement vectors.

• This can be written as the dot product of two vectors:

Types of Work

• Positive Work: When the force has a component in the direction of displacement (), work is positive.

• Negative Work: When the force has a component opposite to the direction of displacement (), work is negative.

• Zero Work: When the force is perpendicular to the displacement (), no work is done.

Examples of Work

• Holding a Barbell Stationary: No work is done because the displacement is zero.

• Lowering a Barbell: The barbell does positive work on the weightlifter's hands (force and displacement in the same direction), while the weightlifter's hands do negative work on the barbell (force and displacement in opposite directions).

Kinetic Energy

Definition and Properties

Kinetic energy is the energy of motion of a particle. It is a scalar quantity and depends only on the mass and speed of the object, not the direction of motion.

• Formula:

• Kinetic energy is always non-negative and is zero only when the object is at rest.

• The SI unit of kinetic energy is the joule (J).

• Kinetic energy increases linearly with mass and with the square of speed.

• Example: Two objects with the same mass and speed but moving in different directions have the same kinetic energy.

• Example: Doubling the mass at constant speed doubles the kinetic energy; doubling the speed at constant mass quadruples the kinetic energy.

The Work-Energy Theorem

The work-energy theorem states that the work done by the net force on a particle equals the change in the particle's kinetic energy.

• If , the particle speeds up.

• If , the particle slows down.

• If , the speed remains constant.

Work and Kinetic Energy in Composite Systems

For systems that cannot be represented as a single point mass (e.g., a skater), the work done by external forces may not account for all changes in kinetic energy. In such cases, internal forces and the distribution of mass must be considered.

Additional info: In extended bodies, rotational kinetic energy and internal energy changes may also be relevant.

Work and Energy with Varying Forces

Non-Constant Forces

When forces are not constant, the work done as a particle moves from to is calculated by integrating the force over the path:

• The work is represented by the area under the force vs. position graph between and .

• For a constant force, this area is a rectangle; for a variable force, it may be more complex.

Example: Stretching a Spring

• The force required to stretch a spring a distance is proportional to (Hooke's Law):

• The work done to stretch the spring a distance is:

Work-Energy Theorem for Curved Paths

When a particle moves along a curved path under a varying force, the work done is found by integrating the force along the path:

Power

Definition and Units

Power is the rate at which work is done.

• Average power:

• Instantaneous power:

• The SI unit of power is the watt (W), where .

• Another common unit is the horsepower (hp), where .

Power in Terms of Force and Velocity

In mechanics, power can also be expressed as the dot product of force and velocity:

• This formula gives the instantaneous power delivered by a force acting on a moving object.

Examples of Power

• Lifting a box slowly (over 5 s): If 1000 J of work is done, the average power is .

• Lifting the same box quickly (over 1 s): The average power is .