Work and Kinetic Energy

Physical Quantities and Energy

This chapter introduces the concept of work as a measure of energy transfer due to a force acting on an object as it moves. Work is a scalar quantity and is closely related to kinetic energy, which is the energy of motion. The work-energy theorem is a central result, connecting the total work done by all forces to the change in an object's kinetic energy.

• Work: A type of energy transfer resulting from a force causing displacement.

• Kinetic Energy: The energy an object possesses due to its motion, defined as .

• Work-Energy Theorem: The net work done by all forces on an object equals the change in its kinetic energy: .

• Power: The rate at which work is done, .

Definition of Work

Work is defined for a constant force acting on an object as it moves through a displacement. If the force is parallel to the displacement, the work done is:

• Formula:

• SI Unit: Joule (J), where

• Example: If a force of 5000 N moves an object 20 m,

Work by a Non-Parallel Force

If the force is not parallel to the displacement, only the component of the force in the direction of displacement does work. The general definition uses the dot product:

• Formula:

• is the angle between the force and displacement vectors.

• Scalar Quantity: Work is a scalar, not a vector.

Sign of Work

The sign of work depends on the direction of the force component relative to the displacement:

• Positive Work: If the force component is in the same direction as displacement (, ).

• Negative Work: If the force component is opposite to displacement (, ).

• Zero Work: If force and displacement are perpendicular (, ). Perpendicular forces do no work.

Net Work and Multiple Forces

When multiple forces act on an object, the total (net) work is the sum of the work done by each force, or equivalently, the work done by the net force:

• Formula:

• Alternatively,

Example: Tractor Pulling a Sled

A tractor pulls a sled with a force at an angle above the horizontal, while friction acts opposite to the displacement. The total work can be calculated by summing the work done by each force or by using the net force.

• Work by Tractor:

• Work by Friction:

• Total Work:

Work-Energy Theorem

The work-energy theorem relates the net work done on an object to its change in kinetic energy:

• Formula:

• This is a reformulation of Newton's 2nd law in terms of energy.

• Application: If , the object speeds up; if , it slows down; if , speed does not change.

Work Done by a Spring (Elastic Force)

For a spring obeying Hooke's law, the force is proportional to displacement from equilibrium:

• Hooke's Law:

• Work Done ON the Spring: (from to )

• Work Done BY the Spring:

• Work is calculated as the area under the force vs. position graph.

Power

Power is the rate at which work is done. Average power is defined as:

• Average Power:

• Instantaneous Power:

• Unit: Watt (W), where

• Example: Lifting a box of mass a height in time requires average power

Summary Table: Work in Different Situations

Key Takeaways

• Work is a scalar quantity representing energy transfer due to force and displacement.

• The work-energy theorem provides a powerful method for solving problems involving forces and motion.

• Power quantifies how quickly work is done.

• Forces perpendicular to displacement do no work.

• Spring forces require integration or area-under-curve methods to calculate work.

Additional info: Some context and examples were expanded for clarity and completeness, including the summary table and explicit formulas for spring work and power.