Work and Energy: Study Notes for College Physics

Work and Energy

Introduction to Energy and Kinetic Energy

Energy is a fundamental physical quantity that objects possess. While its effects are observable, its nature is defined by its ability to do work or produce change. The SI unit of energy is the Joule (J).

• Forms of Energy: Energy exists in various forms, including kinetic, potential, thermal, chemical, and nuclear energy.

• Kinetic Energy (KE): The energy an object has due to its motion.

Formula for Kinetic Energy:

$KE = \frac{1}{2}mv^2$

• m = mass (kg)

• v = velocity (m/s)

Example: Calculate the kinetic energy of a 5 kg box moving at 2 m/s.

$KE = \frac{1}{2} \times 5 \times (2)^2 = 10\ J$

Work Done by a Constant Force

Work is the transfer of energy that occurs when a force acts on an object and causes displacement. The SI unit of work is the Joule (J).

• Formula for Work:

$W = Fd\cos\theta$

• F = magnitude of the force (N)

• d = displacement (m)

• \theta = angle between force and displacement

Sign of Work: If the force gives energy to the object, work is positive; if it takes energy away, work is negative.

Example: You pull a 50 kg box vertically up with a constant 100 N force for 2 m. Work done:

$W = 100 \times 2 \times \cos(0^\circ) = 200\ J$

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, not the path taken (path independence).

• Formula for Work by Gravity:

$W_{gravity} = mg\Delta y$

• Positive when object moves downward, negative when upward.

Example: A 15 kg book falls 2 m. Work done by gravity:

$W = 15 \times 9.8 \times 2 = 294\ J$

Work by Gravity on Inclined Planes

When objects move along inclined planes, only the vertical component of displacement matters for work done by gravity.

• For a displacement along an incline of length L at angle \theta:

$\Delta y = L\sin\theta$

Use this in the gravity work formula.

Hooke's Law and Springs

Springs exert a restoring force proportional to their displacement from equilibrium, described by Hooke's Law.

• Hooke's Law:

$F_s = -kx$

• k = spring constant (N/m)

• x = displacement from equilibrium (m)

Work Done by a Spring:

$W_{spring} = \frac{1}{2}k(x_i^2 - x_f^2)$

Example: Compressing a spring with k = 500 N/m by 2 m:

$W = \frac{1}{2} \times 500 \times (0^2 - 2^2) = -1000\ J$

Calculating Net Work

The net work done on an object is the sum of the work done by all forces acting on it.

$W_{net} = \sum W_{all\ forces}$

This net work is related to the change in kinetic energy (Work-Energy Theorem).

The Work-Energy Theorem

The Work-Energy Theorem states that the net work done on an object equals its change in kinetic energy.

$W_{net} = \Delta KE = KE_f - KE_i$

Example: If a 4 kg box speeds up from 6 m/s to 10 m/s:

$\Delta KE = \frac{1}{2} \times 4 \times (10^2 - 6^2) = 128\ J$

Calculating Work from Force vs. Displacement Graphs

Work done by a variable force can be found as the area under the force vs. displacement graph.

• Area above the axis = positive work

• Area below the axis = negative work

Introduction to Power

Power is the rate at which work is done or energy is transferred. The SI unit is the Watt (W).

$P = \frac{W}{t}$

• W = work (J)

• t = time (s)

Alternatively, if a constant force moves an object at velocity v:

$P = Fv$

Conservation of Mechanical Energy

Mechanical energy (ME) is the sum of kinetic and potential energy in a system. In the absence of non-conservative forces (like friction), ME is conserved.

$ME = KE + U$

• Potential Energy (U): For gravity, $U = mgh$; for springs, $U = \frac{1}{2}kx^2$.

Conservation of Energy Equation:

$KE_i + U_i = KE_f + U_f$

Example: A 2 kg ball dropped from 100 m: $ME_{top} = ME_{bottom}$

Conservation of Total Energy and Isolated Systems

Total energy is conserved in an isolated system (no external forces). When analyzing energy, define the system and identify if it is isolated.

• Internal forces (like springs) do not change total energy.

• External forces (like friction) can change total energy.

Conservative vs. Non-Conservative Forces

Conservative forces (gravity, springs) store energy that can be fully recovered. Non-conservative forces (friction, air resistance) dissipate energy as heat or other forms.

Conservative Forces	Non-Conservative Forces
Gravity	Friction
Spring (Hooke's Law)	Air Resistance

Mechanical energy is conserved only if all forces are conservative.

Conservation of Energy with Non-Conservative Forces

If non-conservative forces do work, mechanical energy is not conserved, but total energy is. The work done by non-conservative forces equals the change in mechanical energy.

$KE_i + U_i + W_{nc} = KE_f + U_f$

• W_{nc} = work done by non-conservative forces

Elastic (Spring) Potential Energy

Springs store energy when compressed or stretched. This energy is called elastic potential energy.

$U_{spring} = \frac{1}{2}kx^2$

Combine with conservation of energy for systems involving springs.

Solving Curved Path and Connected System Problems

For objects moving along curved paths or systems with multiple objects (e.g., pulleys), apply conservation of energy to the entire system, considering all forms of energy and work done by non-conservative forces.

Projectile Motion with Energy Conservation

Energy conservation can be used to solve projectile motion problems, especially when the launch angle is unknown or when only heights and speeds are needed.

Summary Table: Key Equations

Quantity	Equation
Kinetic Energy	$KE = \frac{1}{2}mv^2$
Work (Constant Force)	$W = Fd\cos\theta$
Work by Gravity	$W_{gravity} = mg\Delta y$
Hooke's Law	$F_s = -kx$
Spring Potential Energy	$U_{spring} = \frac{1}{2}kx^2$
Power	$P = \frac{W}{t}$ or $P = Fv$
Work-Energy Theorem	$W_{net} = \Delta KE$
Conservation of Energy	$KE_i + U_i = KE_f + U_f$
With Non-Conservative Forces	$KE_i + U_i + W_{nc} = KE_f + U_f$

Additional info: These notes include worked examples, conceptual explanations, and problem-solving strategies for all major topics in Work and Energy, as covered in a standard college physics course.