Work and Kinetic Energy: Concepts, Examples, and Applications

Work and Kinetic Energy

Introduction

This topic explores the fundamental concepts of work and kinetic energy in physics. It covers the definitions, mathematical formulations, and applications of these concepts, including calculations involving forces, displacement, and energy transformations in various physical scenarios.

Work

Definition of Work

• Work is done when a force causes a displacement of an object in the direction of the force.

• The mathematical expression for work done by a constant force is: $W = F d_{\parallel}$ where $F$ is the magnitude of the force, and $d_{\parallel}$ is the component of displacement parallel to the force.

• If the force and displacement are at an angle $\theta$: $W = F d \cos\theta$

Work on an Inclined Plane

• When pushing an object up an incline, both the applied force and gravity do work.

• Example: Pushing a bicycle of mass 13 kg up a 25° incline for 275 m with a force of 25 N parallel to the road:

  • Work done by the pushing force: $W = F_{\parallel} d = 25\,\text{N} \times 275\,\text{m} = 6875\,\text{J}$

  • Work done by gravity: $W = -mgh = -(9.8\,\frac{\text{m}}{\text{s}^2})(275\,\text{m})\sin25^\circ \approx -14800\,\text{J}$

Work Done by Different Forces

• Only the component of force in the direction of displacement does work.

• Forces perpendicular to displacement (e.g., normal force on a skater moving horizontally) do no work.

• Example: A skater's normal and weight forces act vertically, but if she moves horizontally, neither does work.

Work in Circular Motion

• When an object moves in a circle at constant speed, the tension in the string is always perpendicular to the displacement.

• Therefore, the work done by the tension over one revolution is zero.

Work Done by Gravity and Other Forces

• When lifting or lowering objects, gravity does negative or positive work depending on the direction of movement.

• Example: Lifting a book from a table (0.70 m high) to a shelf:

  • Work done by hand: $W_{\text{hand}} = +45\,\text{J}$ (if upward)

  • Work done by gravity: $W_{\text{gravity}} = -45\,\text{J}$

  • When lowering, the signs reverse.

• The sign convention: Work is negative if force and displacement are in opposite directions.

Work Done by Multiple Forces

• When multiple forces act (e.g., lowering a piano with ropes), calculate work done by each force separately using $W = F_{\parallel} d$.

• Example Table: (Purpose: To compare work done by different forces in lowering a piano)

  Force	Work (J)
  Weight ($W_w$)	7920
  Tension 1 ($W_{T_1}$)	4580
  Tension 2 ($W_{T_2}$)	4580

  Additional info: The sum of the work done by all forces equals the total change in energy.

Work Done by an Escalator

• When an escalator lifts a person of mass $m$ by height $h$, the work done is: $W = mgh$

• Example: For $m = 60$ kg, $h = 5$ m: $W = (60\,\text{kg})(9.8\,\frac{\text{m}}{\text{s}^2})(5\,\text{m}) = 2940\,\text{J}$

Work Done by a Variable Force

Area Under Force-Displacement Curve

• When force varies with position, work is the area under the force vs. displacement graph.

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

Work Done in Stretching a Spring (Hooke's Law)

• Hooke's Law: $F = -kx$, where $k$ is the spring constant and $x$ is the displacement from equilibrium.

• Work done to stretch a spring from $x=0$ to $x$: $W = \frac{1}{2} k x^2$

• This is the area under the linear force-displacement curve for a spring.

Work-Energy Theorem

Statement and Formula

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

• Mathematically: $W_{\text{total}} = \Delta K$ $\Delta K = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2$

Kinetic Energy

• Kinetic energy ($K$) is the energy associated with the motion of an object: $K = \frac{1}{2} m v^2$ where $m$ is mass and $v$ is velocity.

• Example: A cheetah running at 33 m/s: For $m = 50$ kg (example mass), $K = \frac{1}{2} (50)(33^2) = 27,225$ J

• Additional info: Actual mass of a cheetah may vary; calculation is illustrative.

Power

Definition and Formula

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

• Mathematically: $P = \frac{W}{t}$ where $W$ is work and $t$ is time.

• The SI unit of power is the watt (W): $1\,\text{W} = 1\,\text{J/s}$

• For constant force and velocity: $P = F v$

Examples and Applications

• Example: A 70 kg climber ascends 90 m in 10 minutes: $P = \frac{mgh}{t} = \frac{(70)(9.8)(90)}{600} \approx 103$ W

• Example: A microwave oven delivers 105 W of power to melt ice requiring 33000 J: $t = \frac{33000}{105} \approx 314$ s

Summary Table: Key Formulas

Quantity	Formula	Units
Work (constant force)	$W = F d \cos\theta$	J (joules)
Kinetic Energy	$K = \frac{1}{2} m v^2$	J (joules)
Work-Energy Theorem	$W_{\text{total}} = \Delta K$	J (joules)
Power	$P = \frac{W}{t}$, $P = F v$	W (watts)
Spring Work	$W = \frac{1}{2} k x^2$	J (joules)

Additional info:

• Some values and examples are inferred for completeness and clarity.

• All equations are standard in introductory college physics.