Kinetic Energy and Work: Concepts, Theorems, and Applications

Chapter 2: Kinetic Energy and Work

1. Introduction

This chapter introduces the concepts of energy, kinetic energy, and work, which are foundational to understanding the principles of mechanics in physics. These concepts are essential for analyzing how forces affect the motion and energy of objects.

• Energy: A scalar quantity associated with the state or condition of one or more objects. Energy is the ability to do work.

• Kinetic Energy (K): The energy associated with the state of motion of an object. The faster an object moves, the greater its kinetic energy. When stationary, kinetic energy is zero. $K = \frac{1}{2}mv^2$

• Work (W): Energy transferred to or from an object by means of a force acting on it. - Positive work: Energy transferred to the object. - Negative work: Energy transferred from the object.

Scalar Product (Dot Product)

• The scalar product (or dot product) is a way to combine two vectors to yield a scalar (not a vector).

• It is based on the projection of one vector onto another and measures how closely two vectors align in direction.

• Mathematically: $\vec{A} \cdot \vec{B} = AB \cos\theta$ Where $A$ and $B$ are magnitudes, $\theta$ is the angle between them.

• In component form: $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y$

• Physical application: The calculation of work: $W = \vec{F} \cdot \vec{d} = Fd\cos\theta$

2. Work and Kinetic Energy

Work

Work is done when a force causes displacement of an object. Only the component of force in the direction of displacement does work.

• Formula for work by a constant force: $W = Fd\cos\Phi$ Where $\Phi$ is the angle between the force and displacement vectors.

• Work is a scalar quantity and has SI units of the joule (J): $1\ J = 1\ kg \cdot m^2/s^2 = 1\ N \cdot m$

• Net Work: When multiple forces act, the net work is the sum of the work done by each force, or the work done by the net force.

• Signs of work:

  • Same direction as displacement: $W > 0$

  • Opposite direction: $W < 0$

  • No component along displacement: $W = 0$

Work-Kinetic Energy Theorem

• The work-kinetic energy theorem relates the net work done on an object to its change in kinetic energy:

$\Delta K = K_f - K_i = W$

• Where $K_f$ is the final kinetic energy, $K_i$ is the initial kinetic energy, and $W$ is the net work done.

• Thus, $K_f = K_i + W$

3. Work Done by the Gravitational Force

The work done by gravity depends on the direction of displacement relative to the gravitational force.

• Formula: $W_g = mgd\cos\Phi$

• Rising object: $\Phi = 180^\circ$, $W_g = -mgd$ (gravity does negative work, removing energy from the object)

• Falling object: $\Phi = 0^\circ$, $W_g = mgd$ (gravity does positive work, adding energy to the object)

• Work done in lifting/lowering:

  • During upward displacement, applied force does positive work, gravity does negative work.

  • During downward displacement, gravity does positive work, applied force does negative work.

  • The change in kinetic energy is the sum of the work done by all forces: $\Delta K = K_f - K_i = W_a + W_g$

  • If $K_f = K_i$, then $W_a = -W_g = -mgd\cos\Phi$

4. Work Done by a General Variable Force

When the force varies with position, the work done is found by integrating the force over the path of motion.

• One-dimensional case: $W = \int_{x_i}^{x_f} F(x)\,dx$

• Geometric interpretation: The work is the area under the $F(x)$ vs. $x$ curve between $x_i$ and $x_f$.

• Work-Kinetic Energy Theorem (Variable Force): $W = \int_{x_i}^{x_f} F(x)\,dx = \int_{v_i}^{v_f} m v\,dv = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$

• Three-dimensional case: $w = \int_{r_i}^{r_f} \vec{F} \cdot d\vec{r} = \int_{x_i}^{x_f} F_x dx + \int_{y_i}^{y_f} F_y dy + \int_{z_i}^{z_f} F_z dz$

5. Work Done by a Spring Force

The force exerted by a spring is a variable force described by Hooke's Law. The work done by or against a spring depends on the displacement from its equilibrium position.

• Hooke's Law: $F_s = -kx$ Where $k$ is the spring constant (N/m), $x$ is the displacement from equilibrium, and the negative sign indicates the force is always directed toward equilibrium.

• Work done by a spring: $W_s = \frac{1}{2}k x_i^2 - \frac{1}{2}k x_f^2$

• If the block ends up closer to $x=0$ (relaxed position), $W_s > 0$; if farther, $W_s < 0$; if at the same distance, $W_s = 0$.

• If $x_i = 0$ and final position is $x_f$, then $W_s = -\frac{1}{2}k x_f^2$

• Work done by an applied force against a spring: $W_a = -W_s$

Example Problems and Applications

• Problem 1: Block on an inclined plane with friction and applied force. Calculations include work by gravity, friction, applied force, change in kinetic energy, and final velocity.

• Problem 2: Crate sliding with wind force; calculation of work done and change in kinetic energy.

• Problem 3: Elevator cab falling; work done by gravity and cable, and kinetic energy at the end of the fall.

• Problem 4: Block pulled on a horizontal surface with friction; calculation of speed and optimal angle for applied force.

• Problem 5: Particle acted on by a variable force; calculation of work done and change in speed.

• Problem 6: Block sliding under a piecewise variable force; calculation of kinetic energy and speed at various points.

• Problem 7: Block attached to a spring; calculation of speed at equilibrium and with friction.

• Problem 8: Measuring spring constant and work done by a spring in a vertical setup.

Summary Table: Key Equations

Concept	Equation	Description
Kinetic Energy	$K = \frac{1}{2}mv^2$	Energy of motion
Work (constant force)	$W = Fd\cos\Phi$	Work by a constant force
Work-Kinetic Energy Theorem	$\Delta K = W$	Change in kinetic energy equals net work
Work by Gravity	$W_g = mgd\cos\Phi$	Work done by gravitational force
Work by Variable Force	$W = \int_{x_i}^{x_f} F(x)dx$	Area under force vs. displacement curve
Hooke's Law (Spring Force)	$F_s = -kx$	Force exerted by a spring
Work by Spring	$W_s = \frac{1}{2}k x_i^2 - \frac{1}{2}k x_f^2$	Work done by a spring

Additional info: The notes include worked examples and solutions to reinforce the application of these concepts to real-world and exam-style problems. Integrals and vector analysis are introduced for variable forces and multi-dimensional cases.