Work, Power, and Energy Conservation in Physics

Work & Power

Definition and Concepts

Work and power are fundamental concepts in physics, describing how energy is transferred and the rate at which this transfer occurs.

• Work (W): The product of force and displacement in the direction of the force. It is measured in joules (J).

• Power (P): The rate at which work is done or energy is transferred. It is measured in watts (W).

Key Equations:

• Work: $W = F \cdot d \cdot \cos\theta$

• Power: $P = \frac{W}{t}$

Example: A football player running at 5.0 m/s is stopped in 1 second by a tackler.

• Original Kinetic Energy (KE): $KE = \frac{1}{2}mv^2$

• Average Power Required: $P = \frac{\Delta KE}{\Delta t}$

Conservation of Mechanical Energy

Principle and Applications

The conservation of mechanical energy states that in the absence of non-conservative forces (like friction), the total mechanical energy (kinetic + potential) of a system remains constant.

• Kinetic Energy (KE): $KE = \frac{1}{2}mv^2$

• Potential Energy (PE): $PE = mgh$

• Conservation Equation: $KE_i + PE_i = KE_f + PE_f$

Example: Dropping a ball of mass m from a height y = h. Find the velocity at various heights using energy conservation.

• At $y = 2h/3$, $y = h/3$, and $y = 0$:

• Use $mgh = \frac{1}{2}mv^2$ to solve for $v$ at each height.

Spring Force & Energy Dissipation

Spring Constant and Energy Loss

When an object is stopped by a spring, the spring's constant determines how much force it exerts. Energy can also be lost to non-conservative forces such as friction or deformation.

• Spring Force (Hooke's Law): $F = -kx$

• Spring Potential Energy: $PE_{spring} = \frac{1}{2}kx^2$

• Energy Dissipated: $E_{lost} = KE_{initial} - PE_{spring}$

Example: A subway train is brought to a stop by a spring bumper. Given the speed, mass, and stopping distance, find the spring constant k and the energy lost to dissipative forces.

• Use $KE = \frac{1}{2}mv^2$ and $PE_{spring} = \frac{1}{2}kx^2$ to solve for k.

• Calculate energy lost if $k$ and $x$ are known.

Summary Table: Key Equations and Concepts

Concept	Equation	Units
Work	$W = Fd\cos\theta$	Joules (J)
Power	$P = \frac{W}{t}$	Watts (W)
Kinetic Energy	$KE = \frac{1}{2}mv^2$	Joules (J)
Potential Energy (Gravity)	$PE = mgh$	Joules (J)
Spring Force	$F = -kx$	Newtons (N)
Spring Potential Energy	$PE_{spring} = \frac{1}{2}kx^2$	Joules (J)