Work, Energy and Power

Definition

Work is done when a force acts upon an object to cause a displacement. Scientifically, it is the product of the component of the force in the direction of displacement and the magnitude of the displacement.

Work by a Constant Force

When the force doesn't change its magnitude or direction:

W = F · d · cos(θ)

• F = Magnitude of force
• d = Displacement
• θ = Angle between force and displacement

Key Cases

• θ = 0°: Force and movement are in the same direction (Max positive work: $W = Fd$).
• θ = 90°: Force is perpendicular to movement (Zero work: $W = 0$).
• θ = 180°: Force opposes movement (Negative work: $W = -Fd$).

Work by a Variable Force

If the force changes (like a stretching spring), we find the area under the Force-Displacement (F-x) graph:

W = ∫ F(x) dx

Examples

• Positive Work: A person pulling a trolley forward.
• Zero Work: A satellite orbiting Earth in a circle (force is perpendicular to motion).

Real-Life Use

Mechanical Engineering: Engineers calculate work done by gas pressure in car engines to ensure they have enough power to move the vehicle's mass against friction.

Kinetic Energy (KE)

The energy an object possesses due to its motion.

KE = ½mv²

The Work-Energy Theorem

The Net Work done on an object is equal to the change in its Kinetic Energy:

Wnet = ΔKE = KEfinal - KEinitial

Examples

• A speeding bullet having high KE due to velocity.
• A moving bowling ball transferring KE to pins.

Real-Life Use

Automobile Safety: Engineers design "crumple zones" in cars. By increasing the distance/time over which a car stops (doing work), they reduce the impact force on passengers using the Work-Energy theorem.

Gravitational Potential Energy (GPE)

Stored energy based on an object's position or height:

U = mgh

Conservation of Mechanical Energy

In a system with only conservative forces, total mechanical energy remains constant:

KEi + PEi = KEf + PEf

Examples

• A stretched rubber band (Elastic PE).
• Water held behind a dam (Gravitational PE).

Real-Life Use

Hydroelectric Power: Water stored at high altitudes (GPE) falls to turn turbines (KE), generating electricity through energy conservation.

Real-Life Use

Brake Systems: Friction (non-conservative) does negative work on bicycle wheels, converting KE into heat energy.

Power is the rate at which work is done.

P = W / t  OR  P = F · v

Unit: Watt (W), where $1 W = 1 J/s$

Examples

• Two people climbing stairs: The one running has higher power.
• A 100W bulb converts energy faster than a 40W bulb.

Real-Life Use

Electric Motors: The "Wattage" of a washing machine tells you how much work it can perform per second.

1. Elastic Collision

• Momentum is conserved.
• Kinetic Energy is also conserved.
• Example: Subatomic particles.

2. Inelastic Collision

• Momentum is conserved.
• Kinetic Energy is NOT conserved (lost to heat/sound).
• Perfectly Inelastic: Objects stick together.

Real-Life Use

Automotive Engineering: Car crashes are inelastic. Engineers study how momentum is transferred and energy is absorbed by the frame.

1. Problem: A man pushes a crate with $50\text{ N}$ at $60^\circ$ to the horizontal. The crate moves $10\text{ m}$. Calculate work.
   Solution:
   • Given: $F=50\text{ N}, d=10\text{ m}, \theta=60^\circ$
   • $W = 50 \times 10 \times \cos(60^\circ) = 500 \times 0.5 = 250\text{ J}$


2. Problem: A $1000\text{ kg}$ car at $10\text{ m/s}$ stops. How much work was done by brakes?
   Solution:
   • Given: $m=1000, v_i=10, v_f=0$
   • $W = \Delta KE = \frac{1}{2}(1000)(0^2 - 10^2) = 500 \times (-100) = -50,000\text{ J}$

A rollercoaster starts with a motor doing Work to increase the car's Potential Energy (PE) at the top of the hill. As it plunges, PE converts to Kinetic Energy (KE).

While gravity (a conservative force) drives the descent, friction and air resistance (non-conservative forces) do negative work, converting some mechanical energy into heat and sound. Engineers must balance these energy transformations to ensure the car has enough KE to clear loops safely!

1. If the angle between Force and Displacement is $90^\circ$, work done is:
   a) Max b) Min c) Zero d) Negative
2. The unit of Power is:
   a) Joule b) Newton c) Watt d) Pascal
3. When a body falls freely, its total mechanical energy:
   a) Increases b) Decreases c) Remains constant d) Fluctuates
4. Kinetic energy of a body of mass $m$ moving with velocity $v$ is:
   a) $mv$ b) $mv^2$ c) $\frac{1}{2}mv$ d) $\frac{1}{2}mv^2$
5. Which is a non-conservative force?
   a) Gravity b) Friction c) Electrostatic d) Spring force
6. In an elastic collision, which is conserved?
   a) Only Momentum b) Only KE c) Both d) Neither
7. Work done by a constant force is a ______ quantity.
   a) Scalar b) Vector c) Tensor d) None
8. If velocity is doubled, KE becomes:
   a) 2x b) 4x c) 8x d) Same
9. Potential energy is a function of:
   a) Velocity b) Time c) Position d) Acceleration
10. Power is defined as:
    a) Rate of momentum change b) Rate of work c) Force $\times$ Time d) Force $\times$ Velocity

Answers: 1(c), 2(c), 3(c), 4(d), 5(b), 6(c), 7(a), 8(b), 9(c), 10(b)

1. Q: State and prove the Work-Energy Theorem.
   A: Net work done is $\Delta KE$. $W = F \cdot d = ma \cdot d$. Since $v^2 - u^2 = 2ad$, $ad = (v^2 - u^2)/2$. Thus $W = m(v^2 - u^2)/2 = \frac{1}{2}mv^2 - \frac{1}{2}mu^2$.
2. Q: Distinguish between Elastic and Inelastic collisions.
   A: In elastic collisions, both momentum and KE are conserved. In inelastic collisions, momentum is conserved but KE is lost to heat/sound.
3. Q: A $5\text{ kg}$ body is thrown up at $20\text{ m/s}$. Find PE at highest point ($g=10$).
   A: At peak, $KE=0$. $KE_{initial} = PE_{final}$. $\frac{1}{2}(5)(20)^2 = 1000\text{ J}$.
4. Q: Why is work by friction usually negative?
   A: Friction acts opposite to displacement ($\theta = 180^\circ$), and $\cos(180^\circ) = -1$.
5. Q: Define 1 Watt and 1 kWh.
   A: $1\text{ W} = 1\text{ J/s}$. $1\text{ kWh} = 3.6 \times 10^6\text{ J}$.

• Work: $W = Fd \cos\theta$
• Kinetic Energy: $\frac{1}{2}mv^2$
• Potential Energy: $mgh$
• Power: Rate of doing work ($W/t$)

Mnemonics:

• "C" for Conservative = Conserves energy.
• "N" for Non-conservative = Not conserved (lost to heat).
• "W-F-D-Cos" $\rightarrow$ Work = Force $\times$ Displacement $\times$ Cosine.