Force: Turning Effects, Equilibrium, and Center of Gravity Study Notes

Fundamental Concepts of Force

• Force and Body Rigidity:
  • Rigid Body: When a force is applied to a rigid body, it generally results only in the motion of the body because the distance between its constituent particles does not change.
  • Non-rigid Body: When a force is applied to a non-rigid body, it can cause changes in the body\'s size or shape, in addition to causing motion.
• Mathematical Definition of Force: Force is defined as the rate of change in linear momentum:
  • $\mathbf{F} = \frac{d(mv)}{dt} = \frac{d\mathbf{p}}{dt}$
  • If mass ($m$) is constant, the formula simplifies to $\mathbf{F} = m \times \mathbf{a}$.
• Vector Nature and Units:
  • Force is a vector quantity.
  • S.I. Unit: newton ($N$).
  • Gravitational Unit: kilogram-force ($kgf$), where $1\,kgf = g\,N$ (average value of $g = 9.8\,m/s^2$).

Translational and Rotational Motions

• Linear or Translational Motion:
  • Occurs when a force acts on a stationary rigid body that is free to move.
  • The body starts moving in a straight path in the direction of the applied force.
  • Example: Pushing a ball lying on a floor.
• Rotational Motion:
  • Occurs if the body is pivoted at a point and the force is applied at a suitable point other than the pivot.
  • The force rotates the body about an axis passing through the pivoted point. This effect is known as the turning effect of the force.
  • Example: A wheel pivoted at its centre being pushed tangentially on its rim; a door rotating about hinges when the handle is pushed.

Moment of a Force or Torque

• Definition: The turning effect on the body about an axis is called the moment of force (or torque). It relates to the inability of a force to produce linear motion when a body is fixed/pivoted.
• Factors Affecting Turning:
  1. The magnitude of the force applied ($F$).
  2. The perpendicular distance of the line of action of the force from the axis of rotation (or pivoted point).
• Measurement of Torque: The moment of force is equal to the product of the magnitude of the force and the perpendicular distance from the axis of rotation to the line of action of the force.
  • $\text{Moment of force} = F \times \text{Perpendicular distance (OP)}$
• Maximizing Torque: To produce the maximum turning effect with a given force, the force should be applied such that 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: $\text{dyne} \times \text{cm}$.
  • Gravitational Units: $kgf\,m$ (S.I.) or $gf\,cm$ (C.G.S.).
  • Conversions:
    • $1\,N\,m = 10^5\,\text{dyne} \times 10^2\,\text{cm} = 10^7\,\text{dyne}\,\text{cm}$
    • $1\,kgf\,m = 9.8\,N\,m$
    • $1\,gf\,cm = 980\,\text{dyne}\,\text{cm}$
  • Note on Units: While work/energy is measured in Joules ($J$), torque is never written as Joules because torque is a vector quantity, whereas work is scalar.
• Sign Conventions and Direction:
  • Anticlockwise Moment: Taken as positive ($+$). Its direction is along the axis of rotation outwards.
  • Clockwise Moment: Taken as negative ($-$). Its direction is along the axis of rotation inwards.
  • The direction of rotation depends on both the point of application and the direction of the force.

Practical Examples of Turning Effects

• Opening a Door: The handle is placed at the free end (maximum distance from hinges). Applying force near the hinges requires significantly more force, and force applied directly on the hinge results in zero torque.
• Hand Flour Grinder: The handle is provided near the rim to maximize distance from the central iron pivot, allowing rotation with minimal force.
• Steering Wheel: Forces are applied tangentially. Applying force at different points (e.g., top vs. bottom) can change the sense of rotation without changing the direction of the force itself.
• Bicycle Pedals: A toothed wheel (larger than the rear wheel) is used so that the perpendicular distance from the axle to the pedal is large, making it easier to turn the wheel.
• Spanner (Wrench): A long handle is used to produce a large moment of force with a small effort. Anticlockwise turns loosen the nut; clockwise turns tighten it.

Concept of a Couple

• Definition: A single force cannot produce rotation alone; rotation is produced by a pair of forces called a couple. These are two equal and opposite parallel forces that do not act along the same line.
• Internal Reactions: In many cases, the couple consists of the external force and an equal/opposite reaction force at the pivot. The reaction force has zero moment about the pivot.
• Moment of Couple:
  • Calculated as the product of either force and the perpendicular distance between the lines of action of the two forces (called the couple arm).
  • $\text{Moment of couple} = F \times d$
• Examples of Couples:
  • Turning a water tap.
  • Tightening the cap of an inkpot.
  • Turning a key in a lock.
  • Winding a clock/watch.
  • Driving a bicycle pedal.

Equilibrium of Bodies

• General Definition: A body is in equilibrium when multiple forces act on it but produce no change in its state of rest or of linear or rotational motion.
• Kinds of Equilibrium:
  1. Static Equilibrium: The body remains at rest under the influence of several forces.
     • Example: A book lying on a table (weight balanced by normal reaction); a beam balance in a horizontal position.
  2. Dynamic Equilibrium: The body remains in its state of constant motion (translational or rotational) under the influence of several forces.
     • Example: A raindrop falling with terminal velocity (weight balanced by buoyancy and friction/viscosity); an aeroplane flying at a constant height; motion of planets around the sun.
• Conditions for Equilibrium:
  1. The resultant of all acting forces must be zero ($\sum \mathbf{F} = 0$).
  2. The algebraic sum of moments of all forces about the axis of rotation must be zero ($\sum \text{Torque} = 0$).

Principle of Moments

• Definition: For a body in equilibrium, the sum of anticlockwise moments about the axis of rotation is equal to the sum of clockwise moments about the same axis.
  • $\text{Sum of anticlockwise moments} = \text{Sum of clockwise moments}$
• Verification Experiment:
  • A metre rule is suspended horizontally by its centre ($O$).
  • Weights ($W_1, W_2$) are hung from spring balances ($A, B$) at different distances ($l_1, l_2$).
  • In horizontal equilibrium, it is verified that $W_1 \times l_1 = W_2 \times l_2$.

Centre of Gravity (C.G.)

• Definition: The C.G. of a body is the point about which the algebraic sum of the moments of weights of all constituent particles is zero. The entire weight ($W$) of the body is considered to act at this point.
  • $W = w_1 + w_2 + w_3...$
• Properties of C.G.:
  • The position depends on the shape of the body and the distribution of mass. It changes if the body is deformed (e.g., a straight wire vs. a circular wire).
  • The C.G. does not necessarily lie within the material of the body (e.g., a ring or hollow sphere).
• C.G. of Regular Objects:
  • Rod: Mid-point of the rod.
  • Circular Disc/Ring/Sphere: Geometric centre.
  • Solid/Hollow Cylinder: Mid-point on the axis.
  • Solid Cone: At height $\frac{h}{4}$ from the base on its axis.
  • Hollow Cone: At height $\frac{h}{3}$ from the base on its axis.
  • Triangular Lamina: Intersection of its medians (centroid).
  • Rectangle/Square/Rhombus/Parallelogram: Intersection of diagonals.
• Determining C.G. of an Irregular Lamina:
  • Use the plumb line method: suspend the lamina from three different points on its edge ($a, b, c$).
  • Draw vertical lines along the plumb line for each suspension.
  • The intersection point of these lines is the C.G. ($G$).

Uniform Circular Motion

• Definition: Motion of a particle moving with constant speed in a circular path.
• Characteristics:
  • Speed: Constant.
  • Velocity: Variable, because the direction of motion changes at every point (tangential to the path).
  • Acceleration: Present (accelerated motion) due to the changing velocity.
• Difference from Linear Motion: Uniform linear motion has constant speed, constant velocity, and zero acceleration. Uniform circular motion has constant speed but variable velocity and non-zero acceleration.

Centripetal and Centrifugal Forces

• Centripetal Force:
  • The force directed toward the centre required to maintain circular motion.
  • Examples: Electrostatic attraction in atoms; gravity in planetary motion; tension in a string whirling a stone.
• Centrifugal Force:
  • A fictitious (virtual) force assumed by an observer in a rotating frame, acting away from the centre.
  • It is not the reaction force of centripetal force; action and reaction must act on different bodies.
  • Magnitude: Equal to centripetal force.
  • Experiment: A ball on a merry-go-round. An external observer sees the string tension provide centripetal force. A person on the ride sees the ball stationary because they assume a centrifugal force balances the tension.

Numerical and Practical Problems

• Torque Calculation: A force of $10\,N$ at $30\,cm$ from pivot results in a torque of $10 \times 0.3 = 3\,N\,m$.
• Lever Length Scaling: To open a nut requiring $60\,N\,m$ with only $50\,N$ of force, the handle must be $L = \frac{60}{50} = 1.2\,m$ long.
• Beam Balance Problem: A uniform metre rule (mass $M$) balanced at $60\,m$ with a $10\,g$ mass at the $100\,cm$ end. Since weight acts at $50\,cm$, the principle of moments gives: $M \times (60 - 50) = 10 \times (100 - 60) \Rightarrow M = 40\,g$.
• See-saw Balance: Two children ($30\,kg$ at $2\,m$ and $50\,kg$ at $2.5\,m$) on one side produce an anticlockwise moment of $185\,kgf\,m$. A man of $74\,kg$ must sit at $x = \frac{185}{74} = 2.5\,m$ on the opposite side to balance it.
• Rule with Spring Balance: A rule of mass $20\,kg$ with a weight of $40\,kgf$ at the $40\,cm$ mark, pivoted at $0\,cm$, and supported at $100\,cm$:
  • $\text{Total Clockwise Moment} = (40 \times 40) + (20 \times 50) = 2600\,kgf\,cm$.
  • $\text{Spring Balance Force} (F) = \frac{2600}{100} = 26\,kgf$.