Work, Energy, and Simple Machines

Work Done by a Constant Force

**Work** is defined scientifically as the product of force applied on an object and the displacement of the object in the direction of the applied force.

**Formula:** W = F × s

where:

W = work done (in joules, J)

F = force applied (in newtons, N)

s = displacement in the direction of force (in metres, m)

**SI Unit:** 1 Joule (J) = 1 Newton × 1 metre = 1 N·m = 1 kg·m²·s⁻²

**Key Characteristics of Work:**

Work is a scalar quantity (has magnitude but no direction; can be positive or negative)

Work depends on three factors: magnitude of force, magnitude of displacement, and the relative direction between force and displacement

The force must be constant for this equation to apply directly

**Important Point:** When describing work, always specify which force is doing work and on which object the work is being done.

**Graphical Representation:** On a force-displacement graph, work done equals the area under the curve. For constant force, this forms a rectangle with area = F × s.

When is Work Done Equal to Zero?

Work done is zero in three situations:

**1. No Force Applied (F = 0):** If no force acts on an object, W = 0, even if displacement occurs.

**2. No Displacement (s = 0):** If an object does not move despite force application, W = 0. Example: Pushing against a rigid wall produces no work because the wall does not move. Though muscles consume energy (causing fatigue), scientifically no work is done on the wall.

**3. Force Perpendicular to Displacement:** When force and displacement are perpendicular to each other, the component of force in the direction of displacement is zero, so W = 0. Example: A girl carrying a box while walking applies upward force to support its weight, but displacement is horizontal. Since force and displacement are perpendicular, no work is done by this force on the box.

Positive and Negative Work Done

**Positive Work:** When displacement is in the same direction as the applied force, work is positive.

Example: Pushing a wheelchair forward (both force and motion are in the same direction)

The object gains energy

**Negative Work:** When displacement is opposite to the direction of applied force, work is negative.

Example: A goalkeeper stopping a moving ball applies force opposite to its motion

The object loses energy

**Important Concept:** When two objects interact, both do work on each other. One does positive work while the other does negative work. Example: In a goalkeeper-ball collision, the goalkeeper does negative work on the ball (reducing its kinetic energy), while the ball does positive work on the goalkeeper's hand (transferring energy to it).

**Example Problem:** A goalkeeper's hand moves back 15 cm while applying a force of 200 N opposite to a ball's motion. Work done = F × s = 200 N × (−0.15 m) = −30 J (negative because force opposes displacement).

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The Work-Energy Theorem

**Definition:** The work-energy theorem states that the work done on an object equals the change in its energy.

**Mathematical Form:** Work done on an object = Change in its energy

**Significance:** This theorem establishes the fundamental connection between mechanical work and energy. It applies whether forces are constant or variable, and for single objects or systems.

**Physical Meaning:**

When positive work is done on an object, it gains energy and gains capacity to do further work

When negative work is done on an object, it loses energy

Work is the mechanism by which energy is transferred between objects through mechanical force

**Real-Life Examples:**

1. **Cricket Ball and Wickets:** A fielder does positive work by throwing a cricket ball. The ball gains kinetic energy. When the ball hits the wickets, it transfers this energy to the wickets, causing them to fall.

2. **Flower Pot Falling:** Work done in lifting a pot to height gives it potential energy. When it falls, gravity does positive work on it, converting potential energy to kinetic energy. The moving pot can then do work on objects below it.

3. **Carrom Game:** The moving striker (doing positive work on white coin, increasing its energy) collides with the white coin (which does negative work on striker, decreasing its energy). The white coin then does positive work on the black coin.

**Important Application:** This theorem allows solving complex problems without detailed analysis of every force, especially when forces are not constant. Instead, we calculate total work done and relate it to energy change.

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Forms of Energy

Energy exists in multiple forms, with the ability to convert from one form to another:

**1. Mechanical Energy:** Energy due to motion or position of objects

Directly connected to forces and motion studied in physics

**2. Thermal Energy:** Energy that makes objects warm or hot

Transferred through heat conduction, convection, radiation

**3. Electrical Energy:** Energy related to position or motion of charges

Powers devices like bulbs, motors, heaters

**4. Light Energy:** Energy that allows us to see

Travels as electromagnetic waves

**5. Sound Energy:** Energy of vibrations of air or other molecules

Transferred through waves

**6. Chemical Energy:** Energy stored in chemical bonds between atoms

Found in food, fuels, batteries

**7. Nuclear Energy:** Energy stored in nuclei of atoms

Released in nuclear reactions and fusion (powers the Sun)

**Energy Conversion Examples:**

Electrical energy → Light energy (in bulb)

Electrical energy → Thermal energy (in water heater)

Chemical energy → Mechanical energy (muscles contracting)

Mechanical energy → Sound energy (ringing bell)

Chemical energy → Thermal energy (burning fuel)

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Mechanical Energy

**Definition:** Mechanical energy is the energy possessed by an object due to its motion or position. It has two components: kinetic energy and potential energy.

Kinetic Energy

**Definition:** Kinetic energy is the energy possessed by an object due to its motion. All moving objects possess kinetic energy.

**Derivation of Formula:**

Consider an object of mass m starting from rest and accelerating under constant force F over displacement s, reaching final velocity v.

Using kinematic equation: v² = u² + 2as

When u = 0: v² = 2as, so s = v²/(2a)

Work done by force: W = F × s = ma × s = ma × [v²/(2a)] = ½mv²

By work-energy theorem, this work equals the kinetic energy gained.

**Mathematical Formula:** K = ½mv²

where:

K = kinetic energy (in joules)

m = mass (in kg)

v = velocity (in m/s)

**SI Unit:** Joule (J)

**Characteristics:**

Kinetic energy is a scalar quantity (always positive or zero)

Kinetic energy is zero when object is at rest (v = 0)

Kinetic energy increases with velocity (proportional to v²)

If velocity doubles, kinetic energy becomes 4 times (since K ∝ v²)

If velocity becomes half, kinetic energy becomes ¼ of original

**Relationship with Work:**

When positive work is done on object and velocity increases, kinetic energy increases

When negative work is done on object and velocity decreases, kinetic energy decreases

When no work is done (W = 0), kinetic energy remains constant

**Example Problem 1:** A vehicle's velocity doubles. How does kinetic energy change?

Initial KE = ½mv²

Final KE = ½m(2v)² = 4 × ½mv²

Kinetic energy becomes 4 times the original

**Example Problem 2:** A cricket ball of mass 0.2 kg bowled at 154.8 km/h (43 m/s).

K = ½ × 0.2 × (43)² = ½ × 0.2 × 1849 = 184.9 J

**Example Problem 3:** Aircraft Landing on Carrier

Mass = 15,000 kg, stopping force = 367,500 N, stopping distance = 100 m

Initial kinetic energy = ½ × 15,000 × v²

Final kinetic energy = 0 (comes to rest)

Work done by wire = Force × displacement = 367,500 × (−100) = −36,750,000 J

By work-energy theorem: 0 − ½ × 15,000 × v² = −36,750,000

½ × 15,000 × v² = 36,750,000

v² = 4,900

v = 70 m/s = 252 km/h

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Potential Energy

**Definition:** Potential energy is the energy possessed by an object due to its position or state. It represents stored energy that can be converted to kinetic energy.

**Types:**

**1. Gravitational Potential Energy:**

Energy due to height of object above reference point (usually ground)

Objects at greater heights possess more potential energy

Related to work done against gravity to raise object to that height

**Mathematical Formula:** U = mgh

where:

U = gravitational potential energy (in joules)

m = mass (in kg)

g = acceleration due to gravity (9.8 m/s² or 10 m/s²)

h = height above reference point (in metres)

**SI Unit:** Joule (J)

**Characteristics of Gravitational PE:**

Depends on choice of reference level (usually ground is taken as h = 0)

Always relative to a reference point

Object at same height has same gravitational PE regardless of path taken

Increases with height and mass

**Derivation:** When an object of mass m is slowly lifted to height h, work done against gravity is W = mgh. By work-energy theorem, this work gets stored as potential energy.

**2. Elastic Potential Energy:**

Energy stored in springs, stretched rubber bands, bent rods

Arises from deformation of elastic materials

When deforming force is removed, stored energy is released

Related to spring constant and extent of deformation

**Example:** A stretched spring, compressed spring, or bent bow all possess elastic potential energy.

**Conservation During Vertical Motion:**

When an object falls from height h:

Initial energy: U = mgh (at top, v = 0, so KE = 0)

Final energy: K = ½mv² (at bottom, h = 0, so PE = 0)

By conservation: mgh = ½mv²

This gives: v = √(2gh)

This shows that gravitational potential energy converts completely to kinetic energy.

**Key Concept:** Potential energy is "stored" energy. It has the capacity to do work. When released under appropriate conditions, it gets converted to kinetic energy and can do work on other objects.

**Example Problem:** A 2 kg pot at height of 5 m on a shelf.

Gravitational PE = mgh = 2 × 10 × 5 = 100 J

If it falls to ground (h = 0):

Final KE = ½mv² = 100 J

v = √(100) = 10 m/s

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Conservation of Mechanical Energy

**Statement:** In the absence of non-conservative forces (like friction), the total mechanical energy of an object remains constant.

**Mathematical Form:** E = K + U = constant

Or: K₁ + U₁ = K₂ + U₂

where:

K = kinetic energy

U = potential energy

Subscripts 1 and 2 refer to two different positions/times

**Conditions for Conservation:**

No friction acts on the object

No air resistance

Only conservative forces (like gravity) do work

No external non-conservative forces

**Energy Transformations During Free Fall:**

When object falls from height h with initial velocity u:

**At Height h:** K₁ = ½mu², U₁ = mgh, Total E = ½mu² + mgh

**At Ground (h = 0):** K₂ = ½mv², U₂ = 0, Total E = ½mv²

By conservation: ½mu² + mgh = ½mv²

This gives: v² = u² + 2gh (matches kinematic equation)

**Graphical Understanding:** During fall:

PE decreases as object comes down

KE increases as object speeds up

Total mechanical energy remains constant

At any point: Total E = K + U = constant

**Example Problem:** Ball thrown upward with initial velocity 20 m/s from ground.

Initial: KE = ½m(20)² = 200m, PE = 0, Total = 200m J

At height h = 10 m: PE = mgh = 10mg = 100m, KE = 200m − 100m = 100m, Total = 200m J

At maximum height h_max: KE = 0, PE = 200m, so h_max = 200m/10g = 20 m

When returning to ground: KE = 200m (same as initial), PE = 0

**When Mechanical Energy is NOT Conserved:**

If friction or air resistance acts:

Mechanical energy decreases

Lost energy appears as heat/thermal energy

Total energy (mechanical + thermal) remains conserved by law of conservation of energy

Work done against friction equals decrease in mechanical energy

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Power

**Definition:** Power is the rate at which work is done or energy is transferred. It measures how fast work is being accomplished.

**Mathematical Definition:** P = W/t

where:

P = power (in watts)

W = work done (in joules)

t = time taken (in seconds)

**SI Unit:** Watt (W) = Joule/second (J/s) = kg·m²·s⁻³

**Other Units:**

1 kilowatt (kW) = 1,000 W

1 megawatt (MW) = 10⁶ W

1 horsepower (hp) = 746 W (used for engines)

**Physical Meaning:**

Higher power means work is done faster

Same work done in less time requires more power

Same work done in more time requires less power

**Alternative Form Using Velocity:**

When force and velocity are in same direction:

P = W/t = (F × s)/t = F × (s/t) = F × v

**Formula:** P = F·v

where v is the velocity of object in direction of force.

**Characteristics:**

Power is a scalar quantity

Instantaneous power varies with velocity

Average power is calculated over total time period

**Comparison of Power:**

Two machines doing same work:

Machine doing work faster has greater power

Machine doing work slower has lesser power

**Example Problem 1:** A man lifts 50 kg load to height 4 m in 2 seconds.

Work done = mgh = 50 × 10 × 4 = 2,000 J

Power = W/t = 2,000/2 = 1,000 W = 1 kW

**Example Problem 2:** Same 50 kg load lifted to 4 m in 4 seconds.

Work done = 2,000 J (same)

Power = W/t = 2,000/4 = 500 W (half the previous power)

**Real-Life Context:**

Electric bulb rated "60 W" consumes 60 joules of energy per second

Car engines rated in kilowatts indicate how fast they can do work

Fitness rating (climbing stairs fast requires more power than climbing slowly)

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Simple Machines

**Definition:** A simple machine is a device that changes the direction or magnitude of a force applied to it, making it easier to perform work. Simple machines allow us to do the same work with less effort.

**Basic Types:**

1. Lever

**Definition:** A lever is a rigid bar that rotates about a fixed point called the fulcrum.

**Components:**

**Fulcrum (F):** Fixed point of rotation

**Load (L):** Weight to be lifted (resistance)

**Effort (E):** Force applied to move the load

**Mechanical Advantage (MA):** Ratio of load to effort = L/E

**Classes of Levers:**

**Class I Lever:** Fulcrum between effort and load

Example: Seesaw, crowbar, scissors, claw hammer

Can increase force (if effort arm > load arm)

Can change direction of force

Mechanical advantage > 1 when effort arm > load arm

**Class II Lever:** Load between fulcrum and effort

Example: Wheelbarrow, bottle opener, nutcracker

Always increases force (mechanical advantage > 1)

Effort and load move in same direction

Good for lifting heavy loads with small effort

**Class III Lever:** Effort between fulcrum and load

Example: Tweezers, fishing rod, human forearm

Decreases force (mechanical advantage < 1)

Increases distance/speed of movement

Useful when large movement is needed

**Principle:** Lever works on principle of moments. At equilibrium: Effort × Effort arm = Load × Load arm

**Mechanical Advantage:** MA = Load/Effort = Effort arm/Load arm

2. Pulley

**Definition:** A pulley is a wheel with a grooved rim through which a rope or string passes.

**Types:**

**Fixed Pulley:**

Pulley is attached to a fixed support

Does not move but changes direction of force

No mechanical advantage (MA = 1)

Effort equals load

Example: Water well pulley, flagpole pulley

**Movable Pulley:**

Pulley moves with the load

Provides mechanical advantage (MA = 2)

Effort = Load/2

Less effort required but greater displacement of effort point

Example: Construction equipment, weightlifting systems

**Pulley System (Block and Tackle):**

Combination of fixed and movable pulleys

Can provide large mechanical advantages

Mechanical advantage = Number of supporting rope segments

More pulleys = more mechanical advantage but more rope needed

**Advantage:** Reduces effort needed and changes direction of applied force

3. Inclined Plane

**Definition:** An inclined plane is a flat surface set at an angle to the horizontal, used to raise or lower objects more easily.

**Principle:** Instead of lifting object vertically (requiring large force), it can be moved along incline with smaller effort force.

**Components:**

Length of incline (L)

Height to be raised (h)

Base of incline (b)

Angle of inclination (θ)

**Mechanical Advantage:** MA = Length of incline/Height = L/h

**Force Required:** Effort = (Load × h)/L = Load × sin(θ)

**Advantage:**

Reduces effort by distributing weight over longer distance

Longer incline = smaller effort required

Trade-off: Greater distance to move load

**Real Examples:**

Ramps for wheelchairs and trucks

Staircases (series of inclined planes)

Screw threads (inclined plane wrapped around cylinder)

Wedge (two inclined planes back-to-back)

4. Wedge

**Definition:** A wedge is an inclined plane with two sloping surfaces, used to split objects or raise loads.

**Examples:**

Axe blade (splitting wood)

Knife (cutting)

Door stopper (holding door open)

Nail (fastening objects)

**Mechanism:** Small force applied perpendicular to back converts to large force along sloping faces (by geometry).

5. Screw

**Definition:** A screw is an inclined plane wrapped around a cylinder, used to fasten objects together or lift loads.

**Principle:** Rotation converts to linear motion. Pitch (distance advanced in one complete rotation) determines mechanical advantage.

**Mechanical Advantage:** MA = Circumference/Pitch = 2πr/Pitch

**Examples:**

Wood screws (fastening)

Bolts and nuts (joining)

Jack screw (lifting)

Lid of container

**Advantage:** Very high mechanical advantage due to multiple rotations

6. Wheel and Axle

**Definition:** A wheel and axle is a system where a wheel (large radius) is fixed to an axle (small radius), rotating together about the same axis.

**Mechanical Advantage:** MA = Radius of wheel/Radius of axle

**Examples:**

Steering wheel of car (reduces effort for turning)

Cycle wheels (different sized sprockets for different MA)

Door knob (compared to bolt)

Ferris wheel

**Advantage:**

Increases force if radius of wheel > radius of axle

Can increase speed or distance of movement

Changes distribution of force

Key Principle of Simple Machines

**Ideal Mechanical Advantage:** MA = Effort arm/Load arm = Load/Effort

**Relationship Between Force and Distance:**

Simple machines trade effort for distance

If MA > 1: Less effort needed but must move through greater distance

If MA < 1: More effort needed but load moves larger distance

Work done (in ideal case) = Force × Distance remains constant

**Real vs. Ideal:**

**Ideal mechanical advantage:** Calculated assuming no friction

**Actual mechanical advantage:** Always less than ideal due to friction losses

**Efficiency:** (Actual MA/Ideal MA) × 100%

**Limitations:**

No simple machine can increase both force and distance (would violate energy conservation)

Real machines always have losses due to friction

Perfect simple machine would require zero energy loss

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Energy in Simple Machines

When simple machines are used:

**Work Input:** Work done by applied effort force

W_input = Effort × Distance moved by effort

**Work Output:** Useful work done on load

W_output = Load × Distance moved by load

**In Ideal Machine:** W_input = W_output (no energy loss)

**In Real Machine:** W_output < W_input (some energy lost as heat due to friction)

**Efficiency:** η = (W_output/W_input) × 100%

The mechanical advantage allows us to do useful work more conveniently, though the total energy required remains the same in ideal cases.