Work, Energy, and Simple Machines — Notes

Work Done by a Constant Force

Work is a fundamental concept in physics that quantizes the energy transferred when a force causes an object to move. In everyday language, "work" can mean many things, but in science, it has a precise definition based on force and displacement.

SI Unit of Work

The standard international (SI) unit of work is the joule (J). It is defined as the work done when a force of one newton (N) causes a displacement of one metre (m) in the direction of the force.

Since $1\text{ N}=1\text{ kg}\cdot {\text{m/s}}^2$, we can also express the joule in base SI units:

When is Work Done Equal to Zero?

From the definition of work done by a constant force, $W=F\times s$, work is zero in three main scenarios:

1. Zero Force ($F=0$): If no force is applied to an object, no work is done on it, regardless of any displacement.
2. Zero Displacement ($s=0$): If an object does not move, even if a force is applied, no work is done. For example, pushing a stationary wall results in zero scientific work, even though your muscles tire. The energy expenditure is internal.
3. Force Perpendicular to Displacement: If the force applied is perpendicular (at ${90}^{\circ }$) to the direction of displacement, the work done is zero. This is because the component of the force acting in the direction of displacement is zero.

• Example: A person carrying a box horizontally while walking applies an upward force to counter gravity. The displacement is horizontal. Since the force and displacement are at ${90}^{\circ }$, no work is done by this upward force on the box.

Positive and Negative Work Done

The sign of the work done depends on the relative direction of the force and the displacement:

• Positive Work: Work done is positive when the applied force and the displacement are in the same direction. This means the force contributes to the motion of the object.
• Example: Pushing a wheelchair forward; the force you apply and the wheelchair's movement are both in the same direction.
• Negative Work: Work done is negative when the applied force and the displacement are in opposite directions. This means the force opposes the motion of the object.
• Example: A goalkeeper stopping a moving ball. The goalkeeper applies a force opposite to the ball's direction of motion, causing it to slow down and stop. The displacement of the ball (while stopping) is opposite to the force applied by the goalkeeper.

Relationship between Force, Displacement, and Work:

Remember

While you might feel tired when holding a heavy object steady, scientifically, no work is done on the object if there is no displacement. The fatigue is due to continuous internal muscle effort.

Exam-smart

Questions might describe scenarios where you need to determine if work is done, and if so, whether it's positive, negative, or zero. For instance, a problem might describe a force pulling a sled up a hill. You'd need to identify the force, the displacement, and their relative directions to calculate the work. If friction is involved, you'd also consider the work done by friction, which is typically negative as it opposes motion.

The Work-Energy Theorem

The Work-Energy Theorem states that the work done on an object is equal to the change in its kinetic energy. When a net force acts on an object, causing it to accelerate, it gains kinetic energy. This energy gain is precisely equal to the work done by that net force.

where:

• $W_{net}$ is the net work done on the object.
• $\mathrm{\Delta }KE$ is the change in the object's kinetic energy.

This theorem implies that:

• If positive work is done on an object, its kinetic energy increases.
• If negative work is done on an object, its kinetic energy decreases.
• If zero net work is done on an object, its kinetic energy remains constant (it does not accelerate).

The SI unit of energy is the joule (J), the same as the unit of work.

Remember

Work is a way to transfer energy. Positive work done on an object increases its energy, while negative work done on it decreases its energy.

Exam-smart

This theorem is crucial for problems involving changes in speed. If you are given information about forces acting on an object and its initial and final speeds, you can use the work-energy theorem to find the displacement, or vice versa. For example, a problem might state that a force of 10 N acts on a 2 kg mass, moving it from rest to a speed of 5 m/s. You can calculate the work done and then the displacement.

Forms of Energy

Energy is defined as the capacity to do work. While work and mechanical energy are directly related to forces and motion, energy exists in various forms. It's important to understand that energy can be transformed from one form to another.

Here are some common forms of energy:

• Mechanical Energy: This is the energy associated with the motion and position of an object. It is the sum of kinetic and potential energy.
• Thermal Energy: This is the energy that makes objects warm or hot. It is related to the internal motion of atoms and molecules within a substance. For example, an electric water heater converts electrical energy into thermal energy to heat water.
• Chemical Energy: This is energy stored in the bonds of chemical compounds. When these compounds react, the stored energy is released. The food we eat contains chemical energy, which our bodies convert into mechanical energy for movement. Batteries also store chemical energy.
• Electrical Energy: This is the energy associated with the flow of electric charge (electrons). This is the energy that powers most of our appliances. For instance, a light bulb converts electrical energy into light energy and some thermal energy.
• Light Energy: This is energy that we can see, transmitted as electromagnetic waves. The sun is a primary source of light energy.
• Sound Energy: This is energy produced by vibrations that travel through a medium (like air) as waves. A ringing bell converts mechanical energy into sound energy.
• Nuclear Energy: This is energy stored in the nucleus of an atom. It is released during nuclear reactions like fission or fusion.
• Radiant Energy: This is energy that travels in the form of electromagnetic waves, such as radio waves, microwaves, X-rays, and gamma rays. Light energy is a type of radiant energy.

The NCERT textbook also mentions energy due to motion (which is kinetic energy) and energy due to position (which is potential energy) as components of mechanical energy. These will be discussed in detail in the section on Mechanical Energy.

Energy Transformation

A key principle is that energy is conserved, meaning it cannot be created or destroyed, only changed from one form to another.

Here are some examples of energy transformations:

• Food to Movement: Chemical energy in food is converted into mechanical energy by our muscles.
• Electric Bulb: Electrical energy is converted into light energy and thermal energy.
• Electric Water Heater: Electrical energy is converted into thermal energy.
• Ringing Bell: Mechanical energy is converted into sound energy.
• Bicycle Pedaling: Muscular energy (chemical energy from food converted to mechanical energy in muscles) is converted into kinetic energy of the bicycle and rider, and some thermal energy due to friction.

Remember

Energy is never lost; it only changes its form. This is the principle of conservation of energy.

Exam-smart

You might be given a scenario (like pedaling a bicycle) and asked to identify the different forms of energy involved and how they transform. Focus on tracing the energy pathway from its initial form to its final forms.

Mechanical Energy

Mechanical energy is the energy possessed by an object due to its motion or position. It is the sum of an object's kinetic energy and potential energy. This energy is directly related to the forces and motion an object experiences.

Kinetic Energy

Kinetic energy (K) is the energy an object possesses due to its motion. Any object that is moving has kinetic energy.

• Derivation of Kinetic Energy: Consider an object of mass $m$ moving with an initial velocity $u$. When a constant force $F$ acts on it, it acquires a final velocity $v$ after travelling a displacement $s$. From kinematics, we have:

Rearranging for displacement $s$:

The work done by the force is $W=F\times s$. Using Newton's second law, $F=ma$.

According to the Work-Energy Theorem, the work done on an object is equal to the change in its kinetic energy. Therefore, $W=\mathrm{\Delta }K=K_{final}-K_{initial}$. If the object starts from rest ($u=0$), then $W=K_{final}$. The kinetic energy of an object of mass $m$ moving with velocity $v$ is:

• Key Points:
• Kinetic energy is a scalar quantity.
• The SI unit of kinetic energy is the joule (J).
• If an object's velocity doubles, its kinetic energy increases by a factor of four ($K\propto v^2$).
• Positive work done on an object increases its kinetic energy.
• Negative work done on an object decreases its kinetic energy.

Remember

Kinetic energy depends on the square of the velocity, so a small change in speed can significantly alter the kinetic energy.

Exam-smart

Case studies might involve calculating the kinetic energy of moving objects like vehicles or projectiles, or determining how velocity changes if kinetic energy is known. For example, you might be given the kinetic energy of a ball and its mass, and asked to find its speed.

Potential Energy

Potential energy (U) is the energy stored in an object or system due to its deformation or its position within a force field (like gravity, elasticity, or electromagnetism). It represents the capacity to do work.

• Examples of Stored Energy:
• A stretched rubber band or compressed spring stores elastic potential energy due to deformation. When released, this stored energy can do work.
• Two magnets placed with their like poles facing each other store magnetic potential energy due to their relative positions. They will repel and move apart if released.
• A book held above the ground stores gravitational potential energy due to its position relative to the Earth.

Gravitational Potential Energy

Gravitational potential energy is the potential energy an object possesses due to its position in a gravitational field. Specifically, it's the energy stored in a system of objects interacting via gravity, like the Earth and a ball.

• Derivation of Gravitational Potential Energy: Consider an object of mass $m$ on the surface of the Earth, where its potential energy is defined as zero. To lift this object to a height $h$ against the gravitational force, an external force equal to $mg$ (where $g$ is the acceleration due to gravity) must be applied. The work done by this applied force is:

According to the work-energy theorem, this work done is stored as potential energy in the object. Thus, the gravitational potential energy ($U$) of an object at height $h$ is:

• Key Points:
• The SI unit of potential energy is the joule (J).
• Gravitational potential energy is relative to a chosen reference level (often the ground).
• The expression $U=mgh$ is valid for heights near the Earth's surface where $g$ is approximately constant.

Remember

Potential energy is stored energy and depends on the object's height (or configuration) relative to a reference point.

Exam-smart

Questions can involve calculating the potential energy of objects at different heights, such as a diver before a jump or a water tank at a certain elevation. You might also need to determine the height an object reaches if its potential energy is known.

Conservation of Mechanical Energy

The conservation of mechanical energy states that the total mechanical energy of an object or system remains constant if only conservative forces (like gravity, elastic force) do work on it, and no non-conservative forces (like friction, air resistance) are present.

• Mechanical Energy (E): The sum of kinetic energy ($K$) and potential energy ($U$).

• How it Works: When an object falls freely under gravity (without air resistance), its potential energy decreases as it loses height, but this lost potential energy is converted into kinetic energy as its speed increases. Similarly, when an object is thrown upwards, its kinetic energy decreases as it loses speed, and this energy is converted into potential energy as it gains height. The total mechanical energy ($K+U$) remains constant throughout the motion.
• Example: A Falling Object
• At the highest point (e.g., height $h$, initial velocity $0$): $K=0$, $U=mgh$. Total Energy $E=0+mgh=mgh$.
• At an intermediate point (e.g., height $h^{\mathrm{\prime }}$, velocity $v^{\mathrm{\prime }}$): $K=\frac{1}{2}m{v}^{\mathrm{\prime }2}$, $U=mg{h}^{\mathrm{\prime }}$. Total Energy $E=\frac{1}{2}m{v}^{\mathrm{\prime }2}+mg{h}^{\mathrm{\prime }}$.
• At the lowest point (e.g., height $0$, final velocity $v$): $K=\frac{1}{2}m{v}^2$, $U=0$. Total Energy $E=\frac{1}{2}m{v}^2+0=\frac{1}{2}m{v}^2$. According to the conservation of mechanical energy, $mgh=\frac{1}{2}m{v}^{\mathrm{\prime }2}+mg{h}^{\mathrm{\prime }}=\frac{1}{2}m{v}^2$.

Remember

Mechanical energy is conserved only when non-conservative forces like friction or air resistance are negligible or absent.

Exam-smart

These problems often involve scenarios like a roller coaster, a pendulum, or an object sliding down a frictionless incline. You'll typically be given information about the object's initial state (height, velocity) and asked to find its final state (velocity at a certain height, or maximum height reached).

Power

Power (P) is defined as the rate at which work is done. It measures how quickly work is performed or energy is transferred.

• Formula: Average Power $P=\frac{\text{Work Done (W)}}{\text{Time Taken (t)}}$

• Key Points:
• If you do the same amount of work in a shorter time, your power output is higher.
• If you do more work in the same amount of time, your power output is higher.
• The SI unit of power is the watt (W).
• 1 watt = 1 joule per second ($1\text{ W}=1\text{ J/s}$).
• Another common unit for power is horsepower (hp), where $1\text{ hp}=746\text{ W}$.

Remember

Power is about the speed of doing work, not the total amount of work done.

Exam-smart

These questions usually involve comparing the power output of different individuals or machines performing tasks. For example, comparing two people lifting the same weight to different heights in different times, or calculating the power of a motor used to lift an object.

Simple Machines

Simple machines are devices that help us perform tasks, especially those involving moving or lifting heavy objects against forces like gravity. While they cannot reduce the total amount of work done, they make the task feel easier by changing the magnitude or direction of the force needed. The force we apply to operate a machine is called the effort, and the force that the machine helps us overcome is called the load.

A key concept to understand how machines help is mechanical advantage (MA). It's defined as the ratio of the load to the effort:

A mechanical advantage greater than 1 means the machine allows you to lift a heavier load with less effort.

Pulley

A pulley is a simple machine consisting of a wheel with a groove around its rim through which a rope or cable passes.

• Fixed Pulley: This type of pulley is attached to a rigid support.
• Function: A fixed pulley primarily changes the direction of the force. For example, it allows you to pull downwards to lift an object upwards, which is often more convenient.
• Mechanical Advantage: In an ideal fixed pulley (where friction is negligible), the effort required is equal to the load. Therefore, its mechanical advantage is 1.

• Application: Commonly seen in flagpoles and wells.

• Movable Pulley: In this type, the pulley itself moves along with the load.

• Function: Movable pulleys can reduce the magnitude of the effort required to lift a load.

• Mechanical Advantage: A single movable pulley can have a mechanical advantage greater than 1, meaning the effort needed is less than the load.

• Application: Used in systems to lift very heavy objects, often in combination with fixed pulleys to form a pulley system.

Remember

A fixed pulley offers convenience by changing the direction of force, not by reducing the magnitude of the force needed. Its mechanical advantage is always 1.

Exam-smart

You might be given a scenario about lifting a heavy object using a rope and pulley. You'll need to identify whether it's a fixed or movable pulley (or a system) based on how the force is applied and how the object moves. If it's a fixed pulley, the effort equals the load. If it's a movable pulley, the effort is less than the load, and you might need to calculate the MA if given load and effort values.

Pulley

A pulley is a simple machine that consists of a wheel on an axle or shaft that is designed to support movement and change of direction of a taut cable or belt, or transfer of power between the shaft and cable or belt. Pulleys are widely used in real life, such as in elevators and cranes, because they make lifting heavy objects easier by changing the direction of the force or by providing a mechanical advantage.

There are two main types of pulleys:

1. Fixed Pulley:

• A fixed pulley has its axle fixed in position, usually attached to a rigid support.
• It changes the direction of the effort applied. If you pull down on the rope, the load is lifted up.
• The effort required is equal to the load (ignoring friction), as the distance the effort moves is equal to the distance the load moves.
• The mechanical advantage (MA) of a fixed pulley is always 1.
• Formula for Mechanical Advantage (MA):

• For a fixed pulley:

• This means Effort = Load.
• Usefulness: While it doesn't reduce the force needed, it makes the task easier by allowing us to pull downwards, which is often more convenient than pulling upwards.

2. Movable Pulley:

• A movable pulley has its axle attached to the load and moves along with it.
• It helps to reduce the effort required to lift a load.
• The effort required is half the load (ignoring friction), because the load is supported by two sections of the rope.
• The mechanical advantage (MA) of a movable pulley is 2.

• This means Effort = $\frac{1}{2}$ Load.
• Trade-off: To lift the load by a certain height, the rope must be pulled by twice that distance. The work done, however, remains the same (ignoring friction).

3. System of Pulleys (Block and Tackle):

• A system of pulleys combines multiple fixed and movable pulleys.
• This arrangement provides a mechanical advantage greater than 1, allowing much heavier objects to be lifted with much smaller effort.
• The mechanical advantage of a system of pulleys is approximately equal to the number of rope segments supporting the load.
• Usefulness: Used in cranes, elevators, and sailing to lift heavy loads.

Remember

In any pulley system, the work done by the effort is equal to the work done on the load (ignoring friction and other losses). Machines do not create energy; they help us apply it more effectively.

Exam-smart

A question might describe a scenario where a load is lifted using a pulley system. You might be given the load (e.g., weight of an object) and asked to find the effort needed, or vice-versa. You could also be asked to calculate the mechanical advantage or determine the type of pulley system used based on the effort and load values.

Inclined Plane

An inclined plane is a flat supporting surface tilted at an angle, with one end higher than the other, used as an aid for raising or lowering a load. It is a simple machine that allows you to move a heavy object to a higher level with less force than lifting it directly, but over a longer distance.

• How it works: Instead of lifting an object vertically against gravity, you push or pull it up the slope. The force required to push the object up the inclined plane is less than its weight, but you have to move it a greater distance (the length of the incline) to reach the same height.
• Mechanical Advantage: The mechanical advantage of an inclined plane is the ratio of the load (weight of the object) to the effort (force applied to push it up the incline).
• Let $m$ be the mass of the object, so the Load = $mg$.
• Let $h$ be the vertical height of the incline.
• Let $L$ be the length of the inclined plane.
• Let $F^{\mathrm{\prime }}$ be the Effort (force applied to push the object up the incline).
• The work done in lifting the object vertically is $mgh$.
• The work done by applying effort $F^{\mathrm{\prime }}$ along the length $L$ is $F^{\mathrm{\prime }}\times L$ (assuming no friction).
• According to the work-energy theorem (ignoring friction), the work done is the same:

• Therefore, the Effort $F^{\mathrm{\prime }}=\frac{mgh}{L}$.
• The Mechanical Advantage (MA) is:

• Key Principle: Since the length $L$ of the inclined plane is always greater than the height $h$, the mechanical advantage $MA=\frac{L}{h}$ is always greater than 1. This means the effort required ($F^{\mathrm{\prime }}$) is less than the load ($mg$).
• Steeper vs. Shallower Inclines:
• A steeper incline (larger angle with the horizontal, $h$ is large relative to $L$) requires more effort but covers a shorter distance. $MA$ is lower.
• A shallower incline (smaller angle with the horizontal, $h$ is small relative to $L$) requires less effort but covers a longer distance. $MA$ is higher.

Remember

An inclined plane reduces the force required but increases the distance over which the force must be applied. The work done remains the same (ideally).

Exam-smart

You might be given the height and length of a ramp and asked to calculate its mechanical advantage. You could also be given a load and effort, and asked to find if the ramp is suitable or to calculate its dimensions. Real-world examples like roads winding up hills are applications of inclined planes.

Lever

A lever is a rigid bar that pivots around a fixed point called a fulcrum. It is used to multiply force or displacement.

• Components of a Lever:

• Fulcrum: The fixed point around which the lever pivots.

• Load: The object or force that needs to be moved or overcome.

• Effort: The force applied to operate the lever.

• Load Arm: The perpendicular distance from the fulcrum to the line of action of the load.

• Effort Arm: The perpendicular distance from the fulcrum to the line of action of the effort.

• Principle of a Lever: A lever works on the principle of moments. For the lever to be balanced (or in equilibrium), the moment of the effort must be equal to the moment of the load.

• Moment = Force × Perpendicular distance from the fulcrum

• Or, using symbols: $F_1\times d_1=F_2\times d_2$ (where $F_1$ is effort, $d_1$ is effort arm, $F_2$ is load, $d_2$ is load arm).

• Mechanical Advantage (MA):

From the principle of moments ($F_1{d}_1=F_2{d}_2$), we get:

So,

• If the Effort Arm is greater than the Load Arm, the MA is greater than 1, meaning a smaller effort can lift a larger load.

• If the Effort Arm is less than the Load Arm, the MA is less than 1, meaning a larger effort is needed to move a smaller load, but the load moves a greater distance.

• If the Effort Arm is equal to the Load Arm, the MA is 1, and the effort equals the load.

• Classes of Levers: Levers are classified into three classes based on the relative positions of the fulcrum, load, and effort.

Remember

A lever can either multiply force (MA > 1) or multiply distance/speed (MA < 1). The work done by the effort is equal to the work done on the load (ignoring friction).

Exam-smart

Questions may present a lever scenario with given distances and one force, asking you to find the other force. You might also be asked to identify the class of lever for a given tool or situation, or explain why a particular class of lever is used for a specific task (e.g., Class II for multiplying force).