Physics: Work, Energy, and Power

Work

1. Definition:

   • Work is done when a force is applied to an object and the object moves in the direction of the force.
   • Formula: $W = F \cdot d \cdot \cos \theta$
     • $W$: Work done.
     • $F$: Force applied.
     • $d$: Displacement.
     • $\theta$: Angle between the force and displacement.

2. Units:

   • SI Unit: Joule ($J$), where $1 \, J = 1 \, N \cdot m$.

3. Types of Work:

   • Positive Work:
     • Force and displacement are in the same direction.
     • Example: Pushing a box forward.
   • Negative Work:
     • Force and displacement are in opposite directions.
     • Example: Friction opposing motion.
   • Zero Work:
     • Force is perpendicular to displacement or there is no displacement.
     • Example: Holding a bag stationary.

4. Work Done by a Variable Force:

   • For a force that varies with displacement: $W = \int F \, dx$

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Energy

1. Definition:

   • Energy is the capacity to do work.

2. Types of Energy:

   • Kinetic Energy (KE):
     • Energy due to motion.
     • Formula: $KE = \frac{1}{2} mv^2$
       • $m$: Mass, $v$: Velocity.
     • SI Unit: Joule ($J$).
   • Potential Energy (PE):
     • Energy due to position or configuration.
     • Formula: $PE = mgh$
       • $m$: Mass, $g$: Acceleration due to gravity, $h$: Height.
     • SI Unit: Joule ($J$).
   • Mechanical Energy:
     • Sum of kinetic and potential energy: $ME = KE + PE$

3. Conservation of Mechanical Energy:

   • In the absence of non-conservative forces (e.g., friction), the total mechanical energy of a system remains constant: $KE_1 + PE_1 = KE_2 + PE_2$

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Power

1. Definition:

   • Power is the rate at which work is done.
   • Formula: $P = \frac{W}{t}$
     • $P$: Power, $W$: Work done, $t$: Time.
   • Alternate Formula: $P = F \cdot v \cdot \cos \theta$
     • $v$: Velocity.

2. Units:

   • SI Unit: Watt ($W$), where $1 \, W = 1 \, J/s$.
   • Larger Units: Kilowatt ($kW$), Megawatt ($MW$).

3. Types of Power:

   • Average Power: $P_{\text{avg}} = \frac{\Delta W}{\Delta t}$
   • Instantaneous Power: $P = \lim_{\Delta t \to 0} \frac{\Delta W}{\Delta t}$

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Work-Energy Theorem

1. Statement:
   • The work done on an object is equal to the change in its kinetic energy: $W = \Delta KE$
   • Derivation: $W = F \cdot d = ma \cdot d$ $W = m \cdot (v^2 - u^2)/2 \quad \Rightarrow \quad W = \Delta KE$

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Practical Examples

1. Example 1: Calculate the work done by a force of $10 \, N$ acting at an angle of $30^\circ$ to move a box by $5 \, m$.

   • Formula: $W = F \cdot d \cdot \cos \theta$
   • Substituting values: $W = 10 \cdot 5 \cdot \cos 30^\circ = 50 \cdot \frac{\sqrt{3}}{2} = 25\sqrt{3} \, J$

2. Example 2: A $2 \, kg$ object is dropped from a height of $10 \, m$. Find its velocity just before hitting the ground, assuming no air resistance.

   • Using conservation of mechanical energy: $PE_{\text{top}} = KE_{\text{bottom}}$ $mgh = \frac{1}{2} mv^2$
     • Cancel $m$ and solve for $v$: $v = \sqrt{2gh} = \sqrt{2 \cdot 9.8 \cdot 10} = \sqrt{196} = 14 \, m/s$

Types of Forces and Work Done

1. Conservative Forces:

   • Work done is independent of the path taken and depends only on the initial and final positions.
   • Examples: Gravitational Force, Elastic Force.
   • Work done by a conservative force: $W = -\Delta PE$

2. Non-Conservative Forces:

   • Work done depends on the path taken.
   • Examples: Friction, Air Resistance.
   • Energy is dissipated as heat or sound.

3. Work Done by Gravitational Force:

   • For an object of mass $m$ moving from height $h_1$ to $h_2$: $W = m g (h_1 - h_2)$

4. Work Done by Friction:

   • Always opposes motion: $W_f = -f_k \cdot d$

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Potential Energy in Springs (Elastic Potential Energy)

1. Hooke’s Law:

   • The force required to compress or extend a spring is proportional to the displacement: $F = -kx$
     • $k$: Spring constant.
     • $x$: Displacement from equilibrium.

2. Potential Energy Stored in a Spring: $PE_{\text{spring}} = \frac{1}{2} k x^2$

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Power in Practical Situations

1. Power of an Engine:

   • The power delivered by an engine is calculated as: $P = \frac{W}{t} \quad \text{or} \quad P = F \cdot v$
   • Example: A car engine delivering $100 \, kW$ of power can perform $100 \, kJ$ of work per second.

2. Efficiency of Power:

   • Efficiency ($\eta$) is the ratio of useful power output to total power input: $\eta = \frac{\text{Useful Power Output}}{\text{Total Power Input}} \cdot 100$

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Collisions

1. Types of Collisions:

   • Elastic Collision:
     • Both momentum and kinetic energy are conserved.
     • Example: Collision between gas molecules.
   • Inelastic Collision:
     • Only momentum is conserved; kinetic energy is not conserved.
     • Example: Car crashes.

2. Equations for a One-Dimensional Elastic Collision:

   • Final velocities of two objects after collision: $v_1 = \frac{(m_1 - m_2)u_1 + 2m_2 u_2}{m_1 + m_2}$ $v_2 = \frac{(m_2 - m_1)u_2 + 2m_1 u_1}{m_1 + m_2}$
     • $u_1, u_2$: Initial velocities.
     • $v_1, v_2$: Final velocities.
     • $m_1, m_2$: Masses of the objects.

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Numerical Examples

1. Example 1: A spring with a spring constant of $200 \, N/m$ is compressed by $0.1 \, m$. Find the potential energy stored in the spring.

   • Formula: $PE_{\text{spring}} = \frac{1}{2} k x^2$
   • Substituting values: $PE_{\text{spring}} = \frac{1}{2} \cdot 200 \cdot (0.1)^2 = \frac{1}{2} \cdot 200 \cdot 0.01 = 1 \, J$

2. Example 2: A $1500 \, kg$ car accelerates uniformly from $0$ to $20 \, m/s$ over $10 \, s$. Find the work done by the engine and its average power.

   • Work Done: $W = KE = \frac{1}{2} m v^2$ $W = \frac{1}{2} \cdot 1500 \cdot 20^2 = 300,000 \, J$
   • Average Power: $P_{\text{avg}} = \frac{W}{t} = \frac{300,000}{10} = 30,000 \, W = 30 \, kW$

Energy Transformations

1. Kinetic and Potential Energy Interchange:

   • In systems like pendulums or free-falling objects, energy continuously transforms between kinetic and potential energy.
   • At the highest point:
     • Potential Energy is maximum, Kinetic Energy is zero.
   • At the lowest point:
     • Kinetic Energy is maximum, Potential Energy is zero.

2. Conservation of Energy in Real-Life Examples:

   • Roller Coaster:
     • At the top of the track: $PE$ is maximum, $KE$ is minimum.
     • At the bottom: $KE$ is maximum, $PE$ is minimum.
   • Free Fall:
     • Total energy remains constant: $PE + KE = \text{constant}$

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Power in Electrical Systems

1. Electric Power:

   • The rate at which electrical energy is converted into another form (e.g., heat, light).
   • Formula: $P = IV$
     • $I$: Current, $V$: Voltage.
   • Alternate Forms: $P = I^2R \quad \text{or} \quad P = \frac{V^2}{R}$

2. Energy Consumption:

   • The energy consumed by an electrical appliance is given by: $E = P \cdot t$
     • Units:
       • $E$: Joules ($J$) in SI.
       • $E$: Kilowatt-hours ($kWh$) in practical usage.

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Work Done in Rotational Motion

1. Work in Rotational Systems:

   • Analogous to linear motion, work done in rotational motion is: $W = \tau \cdot \theta$
     • $\tau$: Torque.
     • $\theta$: Angular displacement (in radians).

2. Power in Rotational Motion:

   • Power in rotational systems is: $P = \tau \cdot \omega$
     • $\omega$: Angular velocity.

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Efficiency of Machines

1. Definition:

   • Efficiency ($\eta$) is the ratio of useful work output to total work input.
   • Formula: $\eta = \frac{\text{Useful Work Output}}{\text{Work Input}} \cdot 100$

2. Example:

   • If a machine produces $80 \, J$ of useful work for $100 \, J$ of energy input: $\eta = \frac{80}{100} \cdot 100 = 80\%$

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Practical Applications of Work, Energy, and Power

1. Lifting Objects:

   • Work required to lift an object: $W = mgh$

2. Driving Vehicles:

   • Power needed to overcome friction and air resistance.

3. Wind Turbines:

   • Kinetic energy of wind is converted into electrical energy.

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Numerical Examples

1. Example 1: A $2 \, kg$ ball is thrown vertically upward with a velocity of $20 \, m/s$. Find the maximum height it reaches.

   • Using conservation of energy: $KE_{\text{initial}} = PE_{\text{max}}$ $\frac{1}{2} mv^2 = mgh$
     • Cancel $m$ and solve for $h$: $h = \frac{v^2}{2g} = \frac{20^2}{2 \cdot 9.8} = \frac{400}{19.6} \approx 20.41 \, m$

2. Example 2: A wind turbine generates $50 \, kW$ of power. If it runs for $5 \, hours$, calculate the total energy generated in $kWh$.

   • Energy: $E = P \cdot t$ $E = 50 \cdot 5 = 250 \, kWh$

3. Example 3: A car engine delivers $120 \, kW$ to move a car at $30 \, m/s$. Find the force exerted by the engine.

   • Formula: $P = F \cdot v \implies F = \frac{P}{v}$ $F = \frac{120,000}{30} = 4,000 \, N$

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Recap: Key Points to Remember

• Work is force applied over a displacement; it can be positive, negative, or zero.
• Energy exists in various forms and is always conserved in isolated systems.
• Power is the rate of doing work and is crucial in understanding the efficiency of systems.
• Applications of these concepts are observed in everyday devices like engines, wind turbines, and electrical appliances.