Work & Kinetic Energy: Structured Study Notes (Young & Freedman, Ch 06)

Work & Kinetic 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 be transferred or transformed between different forms, but it cannot be created or destroyed (the principle of conservation of energy). The SI unit of energy is the Joule (J).

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

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

• Formula: $K = \frac{1}{2}mv^2$

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

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 thus its kinetic energy. The energy gained by the object comes from the work done by the force.

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

• Formula: $W = Fd\cos\theta$ where $\theta$ is the angle between the force and displacement vectors.

• Positive Work: Force in the direction of motion.

• Negative Work: Force opposite to 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 Friction, Weight, and Normal Forces

Friction always opposes motion and thus always does negative work. Weight and normal forces may do zero work if their direction is perpendicular to displacement.

• Work by Friction: $W_{fk} = f_k d \cos\theta$ (always negative)

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

Work Done by Gravity

Gravity, as a force, can do work on objects. The work done by gravity depends only on the change in vertical position (path independence).

• Formula: $W_g = \pm mg\Delta y$ (positive when object moves down, negative when object moves up)

• Example: A book falling from a shelf; a rock thrown upward.

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

Work by Gravity on Inclined Planes

On inclined planes, the angle in $W = Fd\cos\theta$ is always between the force and displacement, not the incline angle itself.

• Formula: $W_g = -mg\Delta y$

• Example: Calculating work done by gravity for objects moving up or down ramps.

Hooke’s Law & Springs

Springs exert a restoring force that opposes deformation, described by Hooke’s Law. The force is proportional to the displacement from the relaxed position.

• Hooke’s Law: $F_s = -kx$

• Spring Constant (k): Measures stiffness; higher k means harder to deform. Units: N/m.

• Restoring Force: Always opposes the direction of deformation.

• Example: Compressing or stretching a spring; calculating force and displacement.

Work Done by Springs

For variable forces, such as springs, work is calculated using the integral of force over displacement. The work done by a spring is negative when compressing or stretching, as the spring resists deformation.

• Work by Spring: $W_s = -\frac{1}{2}k\Delta x^2$

• Work by Applied Force: $W_{FA} = +\frac{1}{2}k\Delta x^2$

• General Formula: $W_{s, A \rightarrow B} = -\frac{1}{2}k(x_B^2 - x_A^2)$

• Example: Compressing a spring from one length to another; calculating additional work required.

Calculating Net Work

The net work done on an object is the sum of the work done by all forces acting on it. This can be calculated either by summing individual works or by using the net force.

• Net Work: $W_{NET} = \sum W = F_{NET}d\cos\theta$

• Example: Pulling a box with multiple forces; calculating net force and net work.

The Work-Energy Theorem

The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy.

• Formula: $W_{NET} = \Delta K = K_f - K_i$

• Application: Used to solve problems where forces are not given but work or energy changes are required.

• Example: Calculating work done to change the speed of a box.

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 from a force-displacement graph for a box.

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: $P_{avg} = \frac{W}{\Delta t} = \frac{E}{\Delta t}$

• Example: Calculating energy usage of a light bulb; average power delivered by a car engine.

Summary Table: Key Formulas

Concept	Formula	Notes
Kinetic Energy	$K = \frac{1}{2}mv^2$	Scalar, always positive
Work (Constant Force)	$W = Fd\cos\theta$	Angle between force and displacement
Work by Gravity	$W_g = \pm mg\Delta y$	Path independent
Work by Spring	$W_s = -\frac{1}{2}k\Delta x^2$	Variable force
Net Work	$W_{NET} = \sum W = F_{NET}d\cos\theta$	Sum of all works
Work-Energy Theorem	$W_{NET} = \Delta K$	Change in kinetic energy
Power	$P_{avg} = \frac{W}{\Delta t}$	Rate of energy transfer

Additional info: Academic context and examples have been expanded for clarity and completeness. Images included are directly relevant to spring mechanics and Hooke's Law.