Work & Energy

Introduction to Energy and Kinetic Energy

Energy is a fundamental physical quantity possessed by objects, though its exact nature is not fully understood. What is known is how energy behaves: it can neither be created nor destroyed, only transformed between different forms. The unit of energy is the Joule (J).

• Forms of Energy: Includes kinetic, potential, thermal, light, sound, electrical, and more.

• Kinetic Energy (KE): The energy due to an object's motion. It is a scalar quantity, always positive, and has no direction.

• Formula:

• Example: Calculate the kinetic energy of a 5 kg box moving at 3 m/s to the right and 2 m/s to the left. Since KE is scalar, direction does not affect the result.

Work Done by a Constant Force

When a constant force is applied to an object at rest, it causes the object to move, increasing its speed and kinetic energy. The energy transferred to the object comes from the work done by the force.

• Work (W): The measure of energy transferred between objects. Work is done on an object when a force causes displacement. Unit: Joule (J).

• Formula: where θ is the angle between the force and displacement vectors.

• Positive Work: Force acts in the direction of motion.

• Negative Work: Force acts against the direction of motion.

• Example: Pulling a 2 kg box with 3 N over 5 m; stopping a 5 kg cart with a 100 N force over 2.5 m.

Work Done by Gravity

Gravity, as a force, can do work on objects. The work done by gravity depends on the direction of motion relative to the force of gravity.

• Formula:

• Positive Work: When an object falls (motion along gravity).

• Negative Work: When an object rises (motion against gravity).

• Path Independence: Work done by gravity depends only on the change in vertical position, not the path taken.

• Example: Calculating work done by gravity for a falling book or a rock thrown upwards.

Calculating Net Work

The net or total work done on an object is the sum of all works done by all forces acting on it. There are multiple ways to calculate net work.

• Net Work Formula:

• Example: Pulling a box with friction; calculating work done by applied force, friction, weight, and normal force.

• Work-Energy Theorem:

Work by Gravity on Inclined Planes

When calculating work on inclined planes, it is important to use the angle between the force and displacement, not the incline angle itself.

• Formula:

• Example: Calculating work done by gravity for a box sliding down or being pulled up an incline.

Hooke’s Law & Springs

Springs exert a restoring force that opposes deformation. The force required to compress or stretch a spring is described by Hooke’s Law.

• Hooke’s Law:

• Spring Constant (k): Measures the stiffness of the spring. Higher k means harder to deform.

• Displacement (x): The change from the spring’s relaxed position.

• Restoring Force: Always acts opposite to the direction of deformation.

• Example: Calculating force and compression for springs with given k and x values.

Work Done by Springs

For variable forces, such as springs, work is calculated using the area under the force vs. displacement graph or by integrating the force over the displacement.

• Work Done by Spring:

• Work Done ON Spring:

• General Formula:

• Example: Calculating work required to compress a spring between two points.

Calculating Work from Force vs. Displacement Graphs

The work done by any force, constant or variable, is equal to the area under the force vs. displacement graph. Areas above the x-axis represent positive work, while areas below represent negative work.

• Area Calculation: Use geometric shapes (rectangles, triangles) to find the area under the curve.

• Example: Calculating work done by a force applied to a box over a distance using the graph.

Introduction to Power

Power is the rate at which work is done or energy is transferred. It is measured in Watts (W), where 1 W = 1 J/s.

• Average Power Formula:

• Example: Calculating energy used by a light bulb per hour; average power delivered by a car engine.

Summary Table: Key Formulas in Work & Energy