Work, Energy, Power, Impulse, and Momentum: Study Notes

Work, Energy, Power, Impulse, and Momentum

Introduction

This study guide covers the fundamental concepts of work, energy, power, impulse, and momentum in classical physics. These topics are essential for understanding how forces cause motion and how energy is transferred and conserved in physical systems.

Work

Definition of Work

Work is done when a force causes a displacement of an object. The amount of work depends on the magnitude of the force, the displacement, and the angle between the force and displacement vectors.

• Mathematical Definition: The work W done by a constant force F over a displacement s is given by the scalar (dot) product:

$W = \vec{F} \cdot \vec{s}$ $W = F s \cos \theta$

• Units: The SI unit of work is the joule (J), where $1\,J = 1\,N \cdot m$.

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

Work Done by a Constant Force

• If the force and displacement are in the same direction ($\theta = 0$): $W = F s$

• If the force is perpendicular to displacement ($\theta = 90^\circ$): $W = 0$

• If the force is opposite to displacement ($\theta = 180^\circ$): $W = -F s$

Sign of Work

• Positive Work: Force has a component in the direction of displacement ($F \cos \theta > 0$).

• Negative Work: Force has a component opposite to displacement ($F \cos \theta < 0$).

• Zero Work: Force is perpendicular to displacement ($F \cos \theta = 0$).

Example: Lifting a Barbell

• When lifting a barbell, the lifter does positive work on the barbell, while gravity does negative work.

• When lowering the barbell, the lifter does negative work, and gravity does positive work.

Work Done by Several Forces

When multiple forces act on an object, the total work is the sum of the work done by each force:

• Algebraic sum: $W_{total} = W_1 + W_2 + ...$

• Or, use the net force: $W = F_{net} s \cos \theta$

Example: Tractor Pulling a Sled

• Given: $F_{tractor} = 5,000\,N$ at $36.9^\circ$ above horizontal, $s = 20\,m$

• Work by tractor: $W_{tractor} = (5,000\,N)(20\,m)\cos(36.9^\circ)$

• Work by friction: $W_{friction} = (3,500\,N)(20\,m)\cos(180^\circ)$

• Total work: sum of all individual works

Kinetic Energy and the Work-Energy Theorem

Kinetic Energy

Kinetic energy ($K$) is the energy of motion:

$K = \frac{1}{2} m v^2$

• Units: Joules (J)

• Scalar quantity

Work-Energy Theorem

The work-energy theorem states that the net work done on a particle equals the change in its kinetic energy:

$W_{net} = \Delta K = K_f - K_i$

• If $W_{net} > 0$, the object speeds up.

• If $W_{net} < 0$, the object slows down.

• If $W_{net} = 0$, the speed remains constant.

Example: Athlete Running

• Find the kinetic energy of an 80-kg athlete running at 10 m/s:

$K = \frac{1}{2} (80\,kg) (10\,m/s)^2 = 4,000\,J$

Work Done by Varying Forces

General Case

When the force varies with position, the work done is the area under the force vs. position graph:

$W = \int_{x_i}^{x_f} F(x) dx$

Work Done by a Spring (Hooke's Law)

• For a spring: $F = -kx$ (Hooke's Law), where $k$ is the spring constant.

• Work done in stretching/compressing a spring from $x_i$ to $x_f$:

$W = \int_{x_i}^{x_f} kx dx = \frac{1}{2}k(x_f^2 - x_i^2)$

• Units of k: N/m

Example: Compressing a Spring

• To stretch a spring by 2 cm requires four times the work needed to stretch it by 1 cm.

Power

Definition of Power

Power is the rate at which work is done or energy is transferred.

• Average Power: $P_{avg} = \frac{\Delta W}{\Delta t}$

• Instantaneous Power: $P = \frac{dW}{dt}$

• Alternative Form: $P = \vec{F} \cdot \vec{v}$

• Units: Watt (W), where $1\,W = 1\,J/s$

Potential Energy

Gravitational Potential Energy

Gravitational potential energy ($U_{grav}$) is the energy associated with an object's position in a gravitational field:

$U_{grav} = mgh$

• Where $m$ is mass, $g$ is acceleration due to gravity, $h$ is height above a reference point.

Conservation of Mechanical Energy

If only conservative forces (like gravity) do work, the total mechanical energy (kinetic + potential) is conserved:

$K_i + U_{i} = K_f + U_{f}$

Example: Throwing a Baseball Upward

• Find the maximum height reached by a 0.145-kg baseball thrown upward at 20.0 m/s (ignoring air resistance):

$\frac{1}{2} m v^2 = mgh$ $h = \frac{v^2}{2g} = \frac{(20.0)^2}{2 \times 9.8} = 20.4\,m$

Momentum and Impulse

Linear Momentum

Momentum ($\vec{p}$) is the product of mass and velocity:

$\vec{p} = m \vec{v}$

• Units: kg·m/s

• Momentum is a vector quantity.

Impulse

Impulse ($\vec{J}$) is the product of the net force and the time interval over which it acts:

$\vec{J} = \vec{F}_{net} \Delta t$

• Impulse is a vector, with units N·s (equivalent to kg·m/s).

Impulse-Momentum Theorem

The impulse delivered to an object equals the change in its momentum:

$\vec{J} = \Delta \vec{p} = m \vec{v}_f - m \vec{v}_i$

Example: Ball Hitting a Wall

• A 0.40-kg ball hits a wall at 30 m/s (left), rebounds at 20 m/s (right):

$\Delta p = m(v_f - v_i) = 0.40(20 - (-30)) = 0.40(50) = 20\,kg\cdot m/s$

Conservation of Momentum

Principle of Conservation

In a closed system with no external forces, the total momentum remains constant:

$\vec{p}_{initial} = \vec{p}_{final}$

Collisions

• Elastic Collision: Both momentum and kinetic energy are conserved.

• Inelastic Collision: Only momentum is conserved; kinetic energy is not.

• Completely Inelastic Collision: Colliding bodies stick together after collision.

Completely Inelastic Collision Equation

$m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2) v_f$

Example: Two Gliders Sticking Together

• After collision, the combined mass moves with a common velocity determined by conservation of momentum.

Summary Table: Key Quantities

Quantity	Definition	Formula	SI Unit
Work (W)	Force applied over displacement	$W = F s \cos \theta$	Joule (J)
Kinetic Energy (K)	Energy of motion	$K = \frac{1}{2} m v^2$	Joule (J)
Power (P)	Rate of doing work	$P = \frac{dW}{dt}$, $P = F v$	Watt (W)
Momentum (p)	Mass times velocity	$p = m v$	kg·m/s
Impulse (J)	Force times time interval	$J = F \Delta t$	N·s
Gravitational Potential Energy (U)	Energy due to position in a gravitational field	$U = mgh$	Joule (J)

Additional info: Some explanations and examples have been expanded for clarity and completeness, and a summary table has been added for quick reference.