Force, Work, Power and Energy - Detailed Notes

Turning Forces and Equilibrium

Syllabus Overview

This section delves into turning forces, moments, equilibrium, and the concept of the center of gravity, with a focus on straightforward examples and direct problem-solving. It provides a foundational understanding essential for more advanced topics in mechanics.

Translational and Rotational Motions: Introduction to both translational (linear) and rotational motions, highlighting their differences and providing a basis for understanding more complex movements.
Moment of a Force (Torque): The turning effect of a force, explained with both C.G.S. and S.I. units, stressing practical applications such as doors, steering wheels, and bicycle pedals to illustrate the concept.
Clockwise and Anticlockwise Moments: A detailed understanding of the direction of moments, crucial for solving equilibrium problems.
Equilibrium Conditions: Covers both translational and rotational equilibrium conditions necessary for a body to be in a state of equilibrium.
Principle of Moments: Practical verification using a meter rule suspended by spring balances with slotted weights, accompanied by simple numerical problems to reinforce understanding.
Center of Gravity: Qualitative understanding with examples of regular and irregular bodies, explaining how the distribution of mass affects the center of gravity.

Introduction to Force

In Class IX, we learned that force applied to a rigid body causes motion, while force on a non-rigid body changes its size, shape, and motion. Mathematically, force is defined as the rate of change of linear momentum:

$F = \frac{d(mv)}{dt} = \frac{dp}{dt}$

If mass $m$ is constant, then $F = ma$ . Force is a vector quantity measured in newtons (N) or kilogram-force (kgf), where $1 \text{ kgf} = g \text{ N}$ , with $g$ being the acceleration due to gravity (approximately $9.8 \text{ m/s}^2$ ). Understanding the relationship between force, mass, and acceleration is fundamental in mechanics.

Moment of a Force and Equilibrium

Translational and Rotational Motions

When a force acts on a rigid body, it can cause:

Linear or Translational Motion: The body moves in a straight path in the direction of the force. This occurs when the force's line of action passes through the center of mass of the body.
Rotational Motion: The body rotates about an axis passing through a pivoted point. This happens when the force's line of action does not pass through the center of mass, creating a torque.

Linear or Translational Motion

When a force acts on a stationary rigid body free to move, it moves in a straight path in the direction of the force. For example, pushing a ball on the floor. The motion continues until an opposing force, like friction, stops it.

Rotational Motion

If the body is pivoted, the force applied at a suitable point rotates the body about the axis through the pivot. This turning effect is rotational motion. For instance, a wheel pivoted at its center rotates when a tangential force is applied. Similarly, a door rotates about its hinges when force is applied to its handle. The position and direction of the applied force greatly influence the ease and direction of rotation.

Moment (Turning Effect) of a Force or Torque

Consider a body pivoted at point O. A horizontal force $F$ is applied, but the body cannot move linearly. Instead, it rotates about the vertical axis through O. This turning effect depends on:

Magnitude of the Force: The amount of force applied. Larger forces produce greater turning effects.
Distance from the Axis of Rotation: The perpendicular distance of the force's line of action from the pivot. The further the force is applied from the pivot, the greater the turning effect.

The turning effect depends on the product of these factors, known as the moment of force or torque.

Measurement of moment of force (or torque):

The moment of a force (or torque) is the product of the magnitude of the force and the perpendicular distance of the line of action of force from the axis of rotation.

$\text{Moment of force} = \text{Force} \times \text{Perpendicular distance}$

In Fig. 1.3:

$\text{Moment of force about point O} = F \times OP$

To produce maximum turning effect, apply the force at a point where the perpendicular distance from the axis of rotation is greatest. This principle is used in many tools and machines.

Units of Moment of Force

The unit of moment of force is the product of the unit of force and the unit of distance.

S.I. unit: newton-metre (Nm)
C.G.S. unit: dyne-cm

If force is measured in gravitational units:

S.I. unit: kgf × m
C.G.S. unit: gf × cm

Relationships:

\begin{aligned}

1 \text{ Nm} &= 10^5 \text{ dyne} \times 10^2 \text{ cm} = 10^7 \text{ dyne cm} \
1 \text{ kgf} \times \text{m} &= 9.8 \text{ Nm} \
1 \text{ gf} \times \text{cm} &= 980 \text{ dyne cm}
\end{aligned}

Clockwise and Anticlockwise Moments

Anticlockwise moment: positive (body turns anticlockwise)
Clockwise moment: negative (body turns clockwise)

The moment of force is a vector quantity. Anticlockwise moment direction is outwards along the axis of rotation, while clockwise is inwards. The sign convention is crucial for analyzing equilibrium in rotational systems.

Factors Affecting Rotation Direction

The direction of rotation depends on:

The point of application of the force: Applying force on different sides of the pivot will result in opposite rotation directions.
The direction of the force: Reversing the direction of the force will reverse the direction of rotation.

Common Examples of Moment of Force

Opening or Shutting a Door: Force is applied normal to the door at the handle, which is at the maximum distance from the hinges. This maximizes the turning effect for a given force.
Hand Flour Grinder: The handle is near the rim to maximize the distance from the center, allowing for easier grinding.
Steering Wheel: Force is applied tangentially on the rim to turn the wheel. Changing the point of application changes the rotation direction. This provides drivers with control over the vehicle's direction.
Bicycle: Force is applied on the foot pedal of a toothed wheel, converting linear motion into rotational motion to drive the bicycle.
Spanner (Wrench): A long handle produces a large moment of force with a small applied force, making it easier to loosen or tighten nuts and bolts.

Conclusion: Turning a body depends on both the magnitude of force and the perpendicular distance from the axis of rotation. Engineers and designers use this principle to create efficient mechanical systems.

Couple

A single force on a pivoted body doesn't cause rotation alone; it requires a pair of forces. The rotation results from the external force and the reaction force at the pivot. The reaction force is equal in magnitude but opposite in direction. This pair of forces is called a couple.

Definition of Couple

Two equal and opposite parallel forces, not acting along the same line, form a couple. A couple is always needed to produce rotation. Unlike a single force, a couple does not produce any translational motion, only rotation.

Examples:

Opening a door
Opening the nut of a car wheel with a wrench
Turning a water tap
Tightening the cap of an inkpot
Turning a key in a lock
Winding a clock
Turning the steering wheel of a car
Driving the pedal of a bicycle

Moment of Couple

Consider a bar AB pivoted at point O. Two equal and opposite forces, each of magnitude F, are applied at ends A and B. The perpendicular distance between the forces is AB = d, known as the couple arm. The forces do not cause translational motion, but each force turns the bar in the same direction, creating a couple that rotates the bar about O.

$\text{Moment of force at A} = F \times OA \text{ (anticlockwise)}$

$\text{Moment of force at B} = F \times OB \text{ (anticlockwise)}$

Total moment of couple:

$\text{Moment of couple} = F \times OA + F \times OB = F \times (OA + OB) = F \times AB = F \times d$

Moment of couple is the product of either force and the perpendicular distance (couple arm) between the forces. The moment of a couple is independent of the point about which moments are taken.

Equilibrium of Bodies

When a force acts on a body, it can cause translational motion if the body is free to move, or rotational motion if fixed at a point. Equilibrium occurs when:

The resultant of all forces is zero. This ensures there is no net force causing translational motion.
The algebraic sum of moments of all forces about the fixed point is zero. This ensures there is no net torque causing rotational motion.

Definition of Equilibrium

When a number of forces acting on a body produce no change in its state of rest or of linear or rotational motion, the body is in equilibrium. In simpler terms, the body is not accelerating in any direction and not rotating.

Kinds of Equilibrium

Static Equilibrium: Body remains at rest under several forces. This is a stable state where the body does not move unless an external force disrupts it.
Dynamic Equilibrium: Body remains in the same state of motion under several forces. The body is moving with constant velocity and zero acceleration.

Examples of Static Equilibrium

A body on a table pulled by equal and opposite forces. The forces balance each other, resulting in no movement.
A book lying on a table, with its weight balanced by the table's reaction force. The weight of the book is equal to the normal force exerted by the table.
A beam balance in a horizontal position, where clockwise and anticlockwise moments are balanced. This is a practical application of the principle of moments.

Examples of Dynamic Equilibrium

A raindrop reaching the Earth with constant velocity, where weight is balanced by buoyant force and air friction. The raindrop no longer accelerates because the forces are balanced.
An airplane moving at constant height, with upward lift balancing its weight. The thrust of the engines balances the drag force.
A stone whirled in a circular path with uniform speed, where string tension provides centripetal force. The tension in the string provides the necessary centripetal force to keep the stone moving in a circle.
Motion of a planet around the sun or satellite around a planet. Gravitational forces maintain their orbits.
Motion of an electron around the nucleus of an atom. Electromagnetic forces maintain the electron's orbit.

Conditions for Equilibrium

The resultant of all forces acting on the body should be zero. Mathematically, this is represented as $\Sigma F = 0$
The algebraic sum of moments of all forces about the point of rotation should be zero. The sum of anticlockwise moments must equal the sum of clockwise moments. Mathematically, this is represented as $\Sigma M = 0$

Principle of Moments

When several forces act on a pivoted body, they rotate it about an axis through the pivot. The resultant moment is the algebraic sum of each force's moment about that point. Anticlockwise moments are positive, and clockwise moments are negative.

Definition

If the algebraic sum of moments of all forces acting on a body about the axis of rotation is zero, the body is in equilibrium.

Sum of anticlockwise moments = Sum of clockwise moments

Physical balances work based on this principle, ensuring accurate measurements.

Verification of the Principle of Moments

Suspend a meter rule horizontally from a fixed support.
Suspend two spring balances with slotted weights on either side of the thread. Adjust weights or positions until the rule is horizontal.

In equilibrium:

$W<em>1 l</em>1 = W<em>2 l</em>2$

Clockwise moment = Anticlockwise moment

Examples

Several solved examples are provided, illustrating the application of the principle of moments in various scenarios. These examples help in understanding how to apply the principle to solve real-world problems.

Centre of Gravity

Definition of Centre of Gravity

The centre of gravity (C.G.) of a body is the point about which the algebraic sum of moments of weights of all the particles constituting the body is zero. The entire weight of the body can be considered to act at this point, howsoever the body is placed.

Key Points

The position of the center of gravity depends on the shape and distribution of mass. Objects with irregular shapes have centers of gravity that are more challenging to determine.
The center of gravity doesn't always have to be within the material of the body, e.g., a ring or hollow sphere. In such cases, the center of gravity is a point in space.
A body can be treated as a point particle of weight W at its center of gravity, simplifying many physics calculations.
Solid cone: $h/4$ from the base.
Hollow cone: $h/3$ from the base.

Centre of Gravity of Regular Objects

Rod: Mid-point
Circular disc: Geometric center
Solid or hollow sphere: Geometric center
Solid or hollow cylinder: Mid-point on the axis
Circular Ring: center of ring
Triangular lamina or scalene triangle: The point of intersection of medians
Parallelogram, rectangular lamina, square or rhombus: The point of intersection of the diagonals

Centre of Gravity and Balance Point

A solid body can be balanced by supporting it at its center of gravity. A uniform meter rule balances at the 50 cm mark because the algebraic sum of moments is zero. Understanding this principle is crucial for designing stable structures.

Determination of Centre of Gravity of Irregular Lamina

The method is to suspend the lamina from different points and draw lines along a plumb line. The intersection of these lines gives the center of gravity. This method is effective for determining the center of gravity of complex shapes.

Uniform Circular Motion

Definition and Characteristics

When a particle moves with a constant speed in a circular path, its motion is called uniform circular motion. Here are the key characteristics:

Constant speed: The magnitude of the velocity remains the same.
Changing direction of motion: The direction of the velocity is continuously changing.
Non-uniform velocity (variable): Because the direction changes, the velocity is not constant.
Accelerated motion: The change in direction implies acceleration.

Direction of Velocity

The direction of velocity at any instant is along the tangent to the circular path at that point, which continuously changes. This tangential velocity is always perpendicular to the radius of the circle.

Difference Between Uniform Circular Motion and Uniform Linear Motion

Feature	Uniform Linear Motion	Uniform Circular Motion

Speed	Constant	Constant
Velocity	Constant	Variable
Acceleration	Zero	Non-zero (accelerated)
Nature of Motion	Unaccelerated	Accelerated

Centripetal and Centrifugal Force

Centripetal Force

Centripetal force is the force required to maintain circular motion, directed towards the center of the circle. The acceleration is variable in direction but constant in magnitude. Without centripetal force, an object would move in a straight line due to inertia.

Examples include:

Electron orbiting the nucleus in an atom (electrostatic force). The positively charged nucleus attracts the negatively charged electron.
Planet orbiting the sun (gravitational force). The sun's gravity keeps the planets in their orbits.
Moon orbiting the Earth (gravitational force). The Earth's gravity keeps the moon in its orbit.
Stone whirled in a circle tied to a string (tension in the string). The tension provides the necessary centripetal force.

Centrifugal Force

Centrifugal force is an outward force experienced by an observer in a rotating frame of reference. It is a fictitious force, not a real force. Centrifugal force is directed away from the center of the circular path. It is perceived due to the inertia of the object.

Experiment to Understand Centrifugal Force

A ball tied to a string on a merry-go-round is observed by two people:

Observer M: standing on the ground outside the merry-go-round
Observer A: standing on the rotating platform of the merry-go-round

Observer M sees the ball moving in a circular path due to the tension in the string providing centripetal force.

Observer A sees the ball as stationary in front of them and explains it by considering the tension in the string balanced by a centrifugal force. For Observer A, the centrifugal force is necessary to explain why the ball isn't moving towards the center.

Conclusion

Centrifugal force is a fictitious force.
Fictitious forces are used to describe motion from a non-inertial (accelerating) frame of reference. These forces are not real but are used to make the physics consistent in the non-inertial frame.