Comprehensive Study Notes on Force, Moments, Equilibrium, and Circular Motion

Syllabus and Fundamental Concepts of Force

The syllabus covers turning forces, the concept of moment of a force, forces in equilibrium, and the centre of gravity through simple examples and numerical problems.
Scope of Syllabus:
- Introduction to translational and rotational motions.
- Moment (turning effect) of a force, also known as torque, including its C.G.S. and S.I. units.
- Common examples: doors, steering wheels, bicycle pedals, etc.
- Clockwise and anticlockwise moments.
- Conditions for translational and rotational equilibrium.
- Principle of moments and its verification using a metre rule.
- Centre of gravity (qualitative only) for regular bodies and irregular laminas.
- Uniform circular motion as an example of constant speed with acceleration.
- Differences between centripetal and centrifugal forces.
Force Definition:
- On a perfectly rigid body, force causes motion.
- On a non-rigid body, force causes motion and changes in size or shape.
- Quantitative definition: Force is the rate of change in linear momentum.
- Formula: $F = \frac{dp}{dt} = \frac{d(mv)}{dt} = ma$ (if mass $m$ is constant).
- Force is a vector quantity.
- S.I. unit: newton ( $N$ ).
- Gravitational unit: kilogram-force ( $kgf$ ), where $1\,kgf = g\,N$ (average $g = 9.8\,m/s^2$ on Earth).

Translational and Rotational Motion

Linear or Translational Motion:
- Occurs when a force acts on a stationary rigid body free to move.
- The body moves in a straight path in the direction of the applied force.
- Example: Pushing a ball lying on a floor.
Rotational Motion:
- Occurs when a body is pivoted at a point and a force is applied at a suitable point.
- The force rotates the body about an axis passing through the pivoted point.
- This is the turning effect of the force.
- Examples:
- A wheel pivoted at its centre with a force applied tangentially on its rim.
- A door rotating about hinges when force is applied to the handle.

Moment of a Force (Torque)

Definition: The moment of a force (or torque) is equal to the product of the magnitude of the force and the perpendicular distance of the line of action of the force from the axis of rotation.
Formula: $M = F \times OP$ , where $OP$ is the perpendicular distance from the pivot $O$ to the line of action of force $F$ .
Factors Affecting Torque:
1. The magnitude of the force applied.
2. The perpendicular distance of the line of action of the force from the axis of rotation (pivoted point).
Maximum Turning Effect: To produce the maximum turning effect with a given force, the force must be applied at a point where the perpendicular distance from the axis of rotation is at its maximum.
Units of Moment of Force:
- S.I. Unit: newton-metre ( $N\,m$ ).
- C.G.S. Unit: dyne-centimetre ( $dyne\,cm$ ).
- Gravitational Units: $kgf\,m$ (S.I.) and $gf\,cm$ (C.G.S.).
- Unit Conversions:
- $1\,N\,m = 10^5\,dyne \times 10^2\,cm = 10^7\,dyne\,cm$
- $1\,kgf\,m = 9.8\,N\,m$
- $1\,gf\,cm = 980\,dyne\,cm$
Important Distinction: The unit $N\,m$ for torque is never written as joule ( $J$ ). Torque is a vector quantity, whereas work or energy (also measured in $N\,m$ ) is a scalar quantity, which is designated as joules.

Direction of Rotation and Convention

Anticlockwise Moment: Conventionally taken as positive ( $+$ ). The direction is along the axis of rotation outwards, towards the observer.
Clockwise Moment: Conventionally taken as negative ( $-$ ). The direction is along the axis of rotation inwards, away from the observer.
Changing Direction of Rotation: Can be achieved by:
1. Changing the point of application of the force.
2. Changing the direction of the force.
Common Examples of Torque application:
- Opening a Door: Handle is placed at the maximum distance from hinges to minimize the force required. Applying force at the hinge results in zero torque ( $d = 0$ ).
- Hand Flour Grinder: Handle is near the rim to allow rotation with small force.
- Steering Wheel: Force is applied tangentially. Reversing direction or changing the point of application (e.g., from point A to point B on the rim) changes the rotation sense.
- Bicycle: Foot pedals drive a larger toothed wheel connected to a smaller rear wheel; the large distance from the axle reduces the required effort.
- Spanner/Wrench: Longer handles allow for loosening tight nuts by providing a large moment with small force.
- Jack Screw: Features a long arm to lift heavy loads with minimal effort.

Couple and Moment of Couple

Definition: A single force cannot cause rotation in a pivoted body; rotation is produced by a pair of forces called a couple. A couple consists of two equal and opposite parallel forces not acting along the same line.
Reaction Force: When applying force to a pivoted body, a reaction force equal and opposite to the applied force is generated at the pivot. The moment of this reaction force about the pivot is zero.
Moment of Couple Formula: $\text{Moment of Couple} = \text{Either Force} \times \text{Perpendicular distance between the two forces}$ .
- $\text{Total Moment} = (F \times OA) + (F \times OB) = F \times (OA + OB) = F \times d$
- The distance $d$ between the two forces is called the "couple arm."
Examples of Couples:
- Using a car wrench with two arms to open a wheel nut.
- Turning a water tap.
- Tightening the cap of an inkpot.
- Turning a key in a lock.
- Winding a clock or watch.

Equilibrium of Bodies

Definition: A body is in equilibrium when several forces acting on it do not change its state of rest or its state of linear or rotational motion.
Kinds of Equilibrium:
1. Static Equilibrium: The body remains at rest under the influence of forces. Examples: A book on a table (weight balanced by reaction), a beam balance in a horizontal position.
2. Dynamic Equilibrium: The body remains in the same state of motion (translational or rotational). Examples: A rain drop falling at terminal velocity (weight balanced by buoyancy and friction), an aeroplane at constant height, a car at constant velocity, a ceiling fan at constant angular velocity, and planets orbiting the sun.
Conditions for Equilibrium:
1. The resultant of all forces acting on the body must be zero (no translational motion).
2. The algebraic sum of moments of all forces about the point of rotation must be zero (no rotational motion).

Principle of Moments

Statement: In equilibrium, the sum of anticlockwise moments is equal to the sum of clockwise moments.
Algebraic Sum: Anticlockwise moments are positive, clockwise are negative. Equilibrium exists if the total sum equals zero.
Verification: Suspend a metre rule at its centre ( $O$ ). Hang weights $W_1$ and $W_2$ at distances $l_1$ and $l_2$ . The rule is in equilibrium when $W_1 \times l_1 = W_2 \times l_2$ .
Physical Balance: A beam balance works on the principle of moments.

Centre of Gravity (C.G.)

Definition: The Centre of Gravity of a body is the point about which the algebraic sum of moments of weights of all constituent particles is zero. The entire weight ( $W = \sum w_i$ ) is considered to act at this point.
Key Characteristics:
- Position depends on the shape and distribution of mass.
- It changes if the body is deformed (e.g., a straight wire bent into a circle).
- It does not have to be within the material (e.g., a ring or hollow sphere).
C.G. Positions for Regular Objects:
- Rod: Mid-point.
- Circular Disc/Ring: Geometric centre.
- Solid/Hollow Sphere: Geometric centre.
- Cylinder: Mid-point on the axis.
- Solid Cone: At height $\frac{h}{4}$ from the base on the axis.
- Hollow Cone: At height $\frac{h}{3}$ from the base on the axis.
- Triangular Lamina: Intersection of medians.
- Parallelogram/Rectangle/Square: Intersection of diagonals.
Stability: For stable equilibrium, the C.G. should be as low as possible, above the base, and near the geometric centre.
Determination for Irregular Lamina: Use the plumb line method. Suspend the lamina from three different holes ( $a$ , $b$ , $c$ ) and draw lines along the plumb line. The point where all lines intersect is the C.G. ( $G$ ).

Uniform Circular Motion

Definition: Motion of a particle with constant speed along a circular path.
Characteristics:
- Speed is uniform, but the direction of motion changes at every point.
- Velocity is variable (non-uniform), making it an accelerated motion.
- Acceleration is present even though speed is constant.
Direction of Velocity: Always tangential to the circular path at any given instant.
Comparison with Linear Motion:
- Uniform Linear Motion: Constant speed, constant velocity, zero acceleration.
- Uniform Circular Motion: Constant speed, variable velocity, non-zero acceleration.

Centripetal and Centrifugal Forces

Centripetal Force:
- Definition: The force acting on a body moving in a circular path, directed towards the centre.
- Necessary for changing the direction of motion.
- Examples: Electrostatic attraction (electrons), gravity (planets/moon), tension in a string (whirling stone).
Centrifugal Force:
- Definition: A fictitious (virtual) force assumed by an observer in a rotating frame, acting away from the centre.
- It has the same magnitude as centripetal force but opposite direction.
- It is not a reaction force to the centripetal force because they appear to act in different frames or contexts.
Merry-go-round Experiment:
- Observer M (on ground): Sees the ball moving in a circle because tension (centripetal force) pulls it toward the centre.
- Observer A (on platform): Sees the ball as stationary. To explain this, A assumes a centrifugal force balances the tension.
- String Breaking: If the string breaks at point $P$ , Observer M sees the ball fly off tangentially ( $PT$ ). Observer A sees the ball move radially away, attributing it to centrifugal force.