👁 Preview — Study, Practice and Revise are open; mock tests and the rest of the syllabus unlock on subscription. Unlock all · ₹4,999
← Back to Physics
Study mode

Work and Machines

Study Practice Revise 🔒 Test

Introduction: Understanding Work and Machines

In everyday language, we often say we have "worked hard" or "done some work" simply meaning we've exerted effort. But in physics, work has a very specific meaning - it is done only when a force causes an object to move in the direction of that force. This precise definition helps us understand how energy is transferred and used in the real world.

Energy is defined as the capacity to do work. Whenever work is done, energy changes forms or transfers from one object to another. This is fundamental in explaining everything from how machines operate to how our bodies perform physical tasks.

Machines are devices designed to make work easier. They may allow us to move heavy loads with less effort or change the direction or magnitude of forces. From simple levers to complex industrial machines, studying how machines work helps us understand their efficiency and practical applications.

In this chapter, we will explore the core ideas behind work, energy, and power, learn about different types of machines, and solve problems to build your confidence and skills for competitive exams.

Work Done

In physics, work done by a force is defined as the product of the magnitude of the force, the displacement of the object, and the cosine of the angle between the force and the displacement directions.

The formula for work done is:

Work Done

\[W = F \times d \times \cos\theta\]

Work done by a force F that moves an object through a displacement d, where \(\theta\) is the angle between force and displacement directions.

W = Work done (joules, J)
F = Force (newtons, N)
d = Displacement (meters, m)
\(\theta\) = Angle between force and displacement

Explanation of terms:

  • Force (F): A push or pull applied on an object, measured in newtons (N).
  • Displacement (d): The distance moved by the object in a specific direction, measured in meters (m).
  • Angle (θ): The angle between the direction of the force and the direction of displacement.

Positive, negative, and zero work:

  • Positive work: When the force has a component in the direction of displacement (θ < 90°), the work done is positive, meaning energy is transferred to the object.
  • Negative work: When the force acts opposite to displacement (θ > 90°), the work done is negative, indicating energy is taken away (e.g., friction slowing down a moving object).
  • Zero work: When the force is perpendicular (θ = 90°) to displacement, no work is done as the force doesn't cause displacement (e.g., carrying a bag while walking horizontally).

Consider a box on the floor being pushed by a force at an angle. The effective force moving the box forward is only the horizontal component of the applied force.

Box d F θ

Energy and Power

Energy is the ability to do work. When work is done on an object, energy changes or is transferred. Two common types of mechanical energy are:

  • Kinetic Energy (KE): Energy possessed by a body due to its motion.
  • Potential Energy (PE): Energy stored in a body due to its position or height.

Power is the rate at which work is done, or energy is transferred, and it measures how quickly work is accomplished.

If \( W \) is work done in time \( t \), then power \( P \) is:

Power

\[P = \frac{W}{t}\]

Power is the work done per unit time.

P = Power (watts, W)
W = Work done (joules, J)
t = Time taken (seconds, s)

Units Comparison:

Quantity Unit Symbol Metric Prefixes Commonly Used
Work / Energy Joule J kJ (kilo), MJ (mega)
Power Watt W kW (kilo), MW (mega)

Simple Machines and Mechanical Advantage

Machines help us do work more easily by changing the size or direction of forces. Simple machines are the basic building blocks for more complex machines. Examples include levers, pulleys, and inclined planes.

The main purpose of a machine is to reduce the effort force needed to do a certain amount of work.

Mechanical Advantage (MA) is a measure of how much a machine multiplies the input effort force to overcome a load. It is defined as:

Mechanical Advantage

\[MA = \frac{\text{Load}}{\text{Effort}}\]

Ratio of output force (load) to input force (effort).

MA = Mechanical Advantage (unitless)
Load = Load force (N)
Effort = Effort force (N)

Velocity Ratio (VR) is the ratio of the distance moved by the effort force to the distance moved by the load.

Velocity Ratio

\[VR = \frac{\text{Distance moved by effort}}{\text{Distance moved by load}}\]

Ratio comparing movements of effort and load.

VR = Velocity Ratio (unitless)

While the velocity ratio tells us the mechanical advantage expected from distances moved, actual mechanical advantage depends on losses like friction.

It is important to note:
MA ≤ VR, since no machine is 100% efficient due to energy losses.

graph TD    IE[Input Effort]    M[Machine]    OE[Output Effort (Load)]    MA[Mechanical Advantage]    IE --> M    M --> OE    IE --> MA    OE --> MA

Worked Examples

Example 1: Calculating Work Done by a Force Easy
A force of 10 N is applied to move a box 5 m along the floor at an angle of 60° to the horizontal. Calculate the work done by the force.

Step 1: Identify given data:

  • Force, \( F = 10\, \text{N} \)
  • Distance, \( d = 5\, \text{m} \)
  • Angle, \( \theta = 60^\circ \)

Step 2: Use work done formula:

\[ W = F \times d \times \cos \theta \]

Step 3: Calculate cosine of 60°:

\[ \cos 60^\circ = 0.5 \]

Step 4: Substitute values and calculate:

\[ W = 10 \times 5 \times 0.5 = 25\, \text{J} \]

Answer: Work done by the force is 25 joules.

Example 2: Mechanical Advantage of a Lever Medium
A lever has an effort arm length of 2 m and a load arm length of 0.5 m. Calculate the mechanical advantage (MA) of the lever.

Step 1: Note given values:

  • Effort arm \( = 2\, \text{m} \)
  • Load arm \( = 0.5\, \text{m} \)

Step 2: Recall that for levers,

\[ MA = \frac{\text{Effort arm}}{\text{Load arm}} \]

Step 3: Substitute values:

\[ MA = \frac{2}{0.5} = 4 \]

Answer: The mechanical advantage of the lever is 4.

Example 3: Power Output of a Machine Easy
A machine does 200 joules of work in 20 seconds. Calculate the power output of the machine.

Step 1: Given data:

  • Work done, \( W = 200\, J \)
  • Time, \( t = 20\, s \)

Step 2: Use power formula:

\[ P = \frac{W}{t} \]

Step 3: Substitute values:

\[ P = \frac{200}{20} = 10\, \text{W} \]

Answer: Power output of the machine is 10 watts.

Example 4: Efficiency Calculation Medium
A machine has an input work of 500 J and delivers an output work of 400 J. Calculate its efficiency.

Step 1: Identify given data:

  • Input work, \( W_{in} = 500\, \text{J} \)
  • Output work, \( W_{out} = 400\, \text{J} \)

Step 2: Use the efficiency formula:

\[ \text{Efficiency} = \frac{W_{out}}{W_{in}} \times 100\% \]

Step 3: Substitute and calculate:

\[ \text{Efficiency} = \frac{400}{500} \times 100\% = 80\% \]

Answer: The machine's efficiency is 80%.

Example 5: Work Done in Lifting a Load with an Inclined Plane Hard
A 100 N load is lifted vertically by 2 m. The same load is lifted using an inclined plane 5 m long. Calculate the work done in both cases.

Step 1: Given data:

  • Load force, \( F = 100\, N \)
  • Vertical height, \( h = 2\, m \)
  • Inclined plane length, \( l = 5\, m \)

Step 2: Calculate work done lifting vertically:

Since force and displacement are in the same direction (θ=0), work done is:

\[ W_{\text{vertical}} = F \times h = 100 \times 2 = 200\, J \]

Step 3: Calculate effort on inclined plane (assuming no friction):

\[ \text{Effort} = F \times \frac{h}{l} = 100 \times \frac{2}{5} = 40\, N \]

Step 4: Calculate work done along inclined plane (force x distance):

\[ W_{\text{inclined}} = \text{Effort} \times l = 40 \times 5 = 200\, J \]

Answer: Work done in both cases is 200 J. The inclined plane reduces effort but increases distance, keeping total work the same, illustrating energy conservation.

Formula Bank

Work Done
\[ W = F \times d \times \cos \theta \]
where: \( W \) = Work done (J), \( F \) = Force (N), \( d \) = Displacement (m), \( \theta \) = angle between force and displacement
Power
\[ P = \frac{W}{t} \]
where: \( P \) = Power (W), \( W \) = Work done (J), \( t \) = Time (s)
Mechanical Advantage (MA)
\[ MA = \frac{\text{Load}}{\text{Effort}} \]
where: Load = load force (N), Effort = effort force (N)
Velocity Ratio (VR)
\[ VR = \frac{\text{Distance effort moves}}{\text{Distance load moves}} \]
unitless ratio of distances moved by effort and load
Efficiency
\[ \text{Efficiency} = \frac{\text{Output work}}{\text{Input work}} \times 100\% \]
where output and input work are in joules (J)

Tips & Tricks

Tip: Always include the cosine of the angle between force and displacement when calculating work.

When to use: Calculating work done by forces applied at an angle.

Tip: Convert all units to SI before calculation, such as centimeters to meters or kilograms to newtons.

When to use: In all numerical problems to maintain unit consistency.

Tip: Use the shortcut "MA = effort arm / load arm" for quick mechanical advantage estimation with levers.

When to use: Lever problems requiring fast calculations.

Tip: Use the efficiency formula to check whether your calculated output and input work values make physical sense.

When to use: Checking answers in machine efficiency problems.

Tip: Draw force-displacement diagrams to help visualize angles, directions, and distances involved in work and machine problems.

When to use: When forces are not aligned along the movement path.

Common Mistakes to Avoid

❌ Ignoring the angle and calculating work as \( W = F \times d \) directly.
✓ Always use \( W = F \times d \times \cos \theta \) including the angle.
Why: Work is only done by the component of the force in the direction of displacement.
❌ Using units like cm or km without converting to meters.
✓ Convert all length measurements to meters before calculating work or power.
Why: SI units ensure consistent and correct calculations.
❌ Confusing mechanical advantage (force ratio) with velocity ratio (distance ratio).
✓ Calculate MA and VR separately and do not assume they are equal.
Why: Due to friction, mechanical advantage is always less than or equal to velocity ratio.
❌ Mixing up input and output work when calculating efficiency.
✓ Clearly identify and use input work as effort work and output work as load work.
Why: Using wrong values leads to incorrect efficiency results.
❌ Calculating power as just work done, without dividing by time.
✓ Compute power using \( P = \frac{W}{t} \) to get watts.
Why: Power measures rate of work over time, not total work done.
✨ AI exam tools — try them free (included in every plan)
Tip: select any text above to Explain / Example / Simplify it.
Curated videos per subtopic
Top YouTube explainers, AI-ranked for your exam and language. Unlocks with subscription.
Unlock

Try Practice next.

Progress tracking is paywalled — subscribe to mark subtopics as understood and save your streak.

Go to practice →
Ask a doubt
Work and Machines · 10 free messages
Ask me anything about this subtopic. You have 10 free messages this session — chat history isn't saved in preview.