Work and Energy Notes

Work and Energy

Chapter Overview

Fundamental principles of work and energy in physics.

Work

Definition of Work: - Work has a very specific meaning in physics.
- Work done on an object by a constant force (both magnitude and direction) is defined as:
- W = F_{\text{ ot}} d - Where:
  - W = Work done,
  - F_{\text{ ot}} = Component of the force parallel to the displacement,
  - d = Magnitude of displacement.

Examples of Work

Moving a Box:
- Frictionless scenario where force is applied.
- Example:
- F = 100 N
- d = 2 m
- Work Calculation: - W = F_{\text{ ot}} \times d = 100N \times 2m = 200J
Carrying an Object:
- Scenario: Carrying a box horizontally with a normal force and gravitational force acting vertically.
- Forces include:
- F_{\text{N}} = m g - Where:
  - m = mass of the object,
  - g = acceleration due to gravity.
- Work done by gravity (W_g) when lifting or moving vertically is zero when the movement is horizontal.
  - Solution: When carrying an object horizontally, the gravitational force acts vertically downwards, perpendicular to the horizontal displacement. Therefore, the work done by gravity is zero \left(Wg=Fgd\cos\left(90\degree\right)=0\right)
Work to Get a Snack:
- Scenario: Picking up a backpack from the floor and walking to a vending machine on the same floor.
- Inquiry: How much net work have you done?
  - Solution: This depends on the specific interpretation.
  - If considering the work done on the backpack to lift it from the floor to carrying height, positive work is done (equal to mgh).
  - If considering the net work done on the backpack while walking horizontally at a constant speed, the net force is zero, thus the net work done on the backpack is zero.
  - If the backpack is picked up, carried, and then returned to the floor, the net work done by gravity over the entire process is zero, and if starting and ending at rest, net work considering all forces would ideally be zero (ignoring internal work by muscles).

True or False: Concepts of Work

A. Only the net force on an object does any work.
B. The component of a force perpendicular to the displacement does positive work.
C. Only the component of a force parallel or anti-parallel to the displacement does work.

Power

Definition of Power: - Power is the rate at which work is done.
- Work does not account for time alone; a constant amount of work occurs regardless of the time taken.
Calculation of Average Power: - P_{\text{avg}} = \frac{W}{t}

Example: Power Calculation

A 3.5 kg box slides 2.0 m along a sticky floor with a kinetic friction force of 17 N acting on it, coming to rest in two seconds. - Inquiry:
- How much work did the frictional force do on the box?
  - Solution: The frictional force opposes the displacement, so the angle between the force and displacement is 180^\circ.
  - Wfriction=Ffriction\times d=-17N\times2.0m=-34J
- What was the magnitude of the average power output of this process?
  - Solution: The magnitude of the average power is the magnitude of the work done divided by the time taken.
  - Pavg=\left\vert\frac{Wfriction}{t}\right\vert=\left\vert\frac{-34J}{2s}\right\vert=17W

Potential Energy

Definition and Characteristics: - Potential energy belongs to a system, not a single object.
- Requires a defined reference point.
- Physically important is the change in potential energy.
- General definition: The change in potential energy is the work required by an external force to move the object without acceleration between two points.

Gravitational Potential Energy

Formula: - PE_g = m g h
- Where:
- PE_g = Gravitational Potential Energy,
- m = mass of the object,
- g = acceleration due to gravity (9.8 \text{ m/s}^2),
- h = height relative to a reference point.
Example: - A 1.0 kg rock is lifted to 2.0 m.
- What is the rock's potential energy relative to its original height?
  - Solution: Assuming original height is h=0.
  - PE_g = m g h = 1.0 \text{ kg} \times 9.8 \text{ m/s}^2 \times 2.0 \text{ m} = 19.6 \text{ J}
- Joseph stood at a height of 50.0 m as he picked up the rock.
- What is the potential energy relative to the base of the cliff?
  - Solution: If the rock is picked up at a height of 50.0 m (relative to the base of the cliff).
  - PE_g = m g h = 1.0 \text{ kg} \times 9.8 \text{ m/s}^2 \times 50.0 \text{ m} = 490 \text{ J}
- When the rock is dropped, calculate its kinetic energy just before hitting the ground.
  - Solution: Assuming the rock is dropped from 50.0 m (the last height mentioned). By conservation of energy, the initial potential energy converts to kinetic energy just before impact (ignoring air resistance).
  - KEf=PEg=490J

Elastic Potential Energy

Definition of Elastic Force: - According to Hooke's Law:
- \left|F\right| = k \left|x\right| - Where:
  - F = force applied to the spring,
  - k = spring constant,
  - x = displacement from the equilibrium position.
Elastic Potential Energy Formula: - PE_{\text{elastic}} = \frac{1}{2} k x^2
- Example of Elastic Potential Energy Calculation:
- Holding a spring stretched to 40.0 cm with a force of 20.0 N. - Calculate the elastic potential energy of the spring.
  - Solution: First, find the spring constant k.
  - k = \frac{F}{x} = \frac{20.0 \text{ N}}{0.40 \text{ m}} = 50.0 \text{ N/m}. (Convert 40.0 \text{ cm} to 0.40 \text{ m}).
  - Now, calculate the elastic potential energy:
  - PE_{\text{elastic}} = \frac{1}{2} k x^2 = \frac{1}{2} \times 50.0 \text{ N/m} \times (0.40 \text{ m})^2 = \frac{1}{2} \times 50.0 \times 0.16 = 4.0 \text{ J}

Electric Potential Energy

Introduction to Electric Potential Energy: - To be revisited in later chapters or exams.

Forces and Energy

Conservative and Nonconservative Forces

Conservative Forces: - Work done by conservative forces depends only on the starting and final positions, not on the path taken.
- Examples: - Gravitational force, Elastic force, Electric force.
Nonconservative Forces: - Work done is path-dependent.
- Examples: - Friction, Air resistance, Tension in cords, Motor or rocket propulsion, Human push/pull.
Important Note: - Potential energy can only be defined for conservative forces.

Work-Energy Theorem

Relationships between Work and Energy: Wnet=Wc+W_{Nc}
- Change in kinetic energy:
- \Delta KE=-\Delta PE+W_{\text{Nc}}
Rearrangement that defines work done: W_{Nc}+KE_{i}+PEg_{i}+PEel_{i}=KEf+PEgf+PEelf
Energy Conservation: - If W_{Nc}=0 then Etotal_{i}=Etotal_{f} ( Energy is conserved).

Applications of Work and Energy

Track Race Scenario

Observation: - Both ramps are nearly frictionless with equal heights y1 and y2
- Inquiry:
- Which ball arrives at the bottom with a greater speed?
- A. The ball on the curved track.
- B. The ball on the straight track.
- C. Both arrive with the same speed.
  - Solution: C. Both arrive with the same speed. Assuming the ramps are frictionless and both balls start from the same height and end at the same height, the total mechanical energy is conserved. Since initial potential energy is the same and converts entirely to kinetic energy, their final kinetic energies, and thus their final speeds, will be identical regardless of the path taken.

Water Slide Calculation

Scenario: - A spring-breaker of mass m is released from rest at a height h = 8.5 m.
- Inquiry:
- Determine the girl's speed at the bottom of the slide assuming it is frictionless.
  - Solution: Using the principle of conservation of mechanical energy \left(KE_{i}+PE_{i}=KEf+PEf\right).
  - Initial state (at rest, height h): Ke_{i}=0,PE_{i}=mgh
  - Final state (at bottom, height 0): KE_{f}=\frac12mv_{f}^2,PE_{f}=0
  - Therefore, mgh = \frac{1}{2}mv_{\text{f}}^2
  - Cancel m: gh = \frac{1}{2}v_{\text{f}}^2
  - Solve for v_{f} : vf=\sqrt{2gh}
  - v_{\text{f}} = \sqrt{2 \times 9.8 \text{ m/s}^2 \times 8.5 \text{ m}} = \sqrt{166.6 \text{ m}^2/\text{s}^2} \approx 12.91 \text{ m/s}

Circus Beagle Example

Scenario: - A circus beagle of mass m = 6.0 kg runs onto the left end of a curved ramp with an initial speed v = 7.8 m/s at height y = 8.5 m.
- After sliding, it comes to a momentary stop at height y = 11.1 m above the floor.
- Inquiry:
- How much work does friction do while the beagle slides, which is equal to the increase in thermal energy due to friction?
  - Solution: Use the Work-Energy Theorem for nonconservative forces: W_{Nc}+KE_{i}+PEg_{i}+PEel_{i}=KEf+PEg_{f}+PEel_{f}
  - Here, W_{Nc} is the work done by \left(Wfriction\right)
  - Initial conditions:
  - m = 6.0 \text{ kg}
  - v_{\text{i}} = 7.8 \text{ m/s}
  - h_{\text{i}} = 8.5 \text{ m}
  - Final conditions:
  - Momentary stop, so v_{\text{f}} = 0 \text{ m/s}
  - h_{\text{f}} = 11.1 \text{ m}
  - KE_{\text{f}} = 0 \text{ J}
  - W_{Nc}=mg\left(h_{f}-h_{i}\right)-\frac12mv^2_{i}=\left(6\right)\left(9.8\right)\left(11.1-8.5\right)-\frac12\left(6\right)\left(7.8\right)^2=-29.64J
  - The increase in thermal energy due to friction is the magnitude of the work done by friction, which is 29.64 \text{ J}.

Formula Sheet

Work Done by a Constant Force:

W = F_{\text{ot}} d

W: Work done on an object (measured in Joules, J).
F_{\text{ot}}: Component of the force parallel to the displacement (measured in Newtons, N).
d: Magnitude of the displacement (measured in meters, m).

Kinetic Energy:

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

KE: Kinetic energy (measured in Joules, J).
m: Mass of the object (measured in kilograms, kg).
v: Speed of the object (measured in meters per second, m/s).

Average Power:

P_{\text{avg}} = \frac{W}{t}

P_{\text{avg}}: Average power (measured in Watts, W).
W: Work done (measured in Joules, J).
t: Time taken to do the work (measured in seconds, s).

Gravitational Potential Energy:

PE_g = m g h

PE_g: Gravitational Potential Energy (measured in Joules, J).
m: Mass of the object (measured in kilograms, kg).
g: Acceleration due to gravity (9.8 \text{ m/s}^2).
h: Height relative to a reference point (measured in meters, m).

Hooke's Law (Elastic Force Magnitude):

\left|F\right| = k \left|x\right|

F: Magnitude of the force applied to the spring (measured in Newtons, N).
k: Spring constant (a measure of the spring's stiffness, measured in Newtons per meter, N/m).
x: Magnitude of the displacement from the equilibrium position (measured in meters, m).

Elastic Potential Energy:

PE_{\text{elastic}} = \frac{1}{2} k x^2

PE_{\text{elastic}}: Elastic Potential Energy (measured in Joules, J).
k: Spring constant (measured in Newtons per meter, N/m).
x: Displacement from the equilibrium position (measured in meters, m).

Work-Energy Theorem (including nonconservative forces):

W_{Nc}+KE_{i}+PEg_{i}+PEel_{i}=KEf+PEg_{f}+PEel_{f}

W_{\text{NC}}: Work done by nonconservative forces (e.g., friction, air resistance) (measured in Joules, J).
KE_{\text{i}}: Initial kinetic energy (J).
PE_{\text{gi}}: Initial gravitational potential energy (J).
PE_{\text{el,i}}: Initial elastic potential energy (J).
KE_{\text{f}}: Final kinetic energy (J).
PE_{\text{gf}}: Final gravitational potential energy (J).
PE_{\text{el,f}}: Final elastic potential energy (J).

Conservation of Mechanical Energy (when W_{\text{NC}} = 0):

Etotal_{i}=Etotal_{f}

Or equivalently:KE_{i}+PE_{i}=KE_{f}+PE_{f}

E_{\text{total,i}}: Initial total mechanical energy (J).
E_{\text{total,f}}: Final total mechanical energy (J).
KE_{\text{i}}: Initial kinetic energy (J).
PE_{\text{i}}: Initial potential energy (gravitational and/or elastic) (J).
KE_{\text{f}}: Final kinetic energy (J).