Physics Chapter 1: Force, Turning Effects, and Circular Motion, and Centre of Gravity

Syllabus Coverage and Introduction to Force

• Syllabus Scope (i): Includes the concept of turning forces, moment of a force, forces in equilibrium, and the centre of gravity (C.G.). It covers elementary introductions to translational and rotational motions, torque and its units (C.G.S. and S.I.), and common examples like a door, steering wheel, and bicycle pedal. It also addresses clockwise and anticlockwise moments, conditions for translational and rotational equilibrium, the principle of moments, and the centre of gravity for regular and irregular bodies.

• Syllabus Scope (ii): Focuses on uniform circular motion as an example of constant speed despite the presence of acceleration (force), and explains the difference between centrifugal and centripetal forces.

• Force on Rigid vs. Non-Rigid Bodies: A force applied to a rigid body causes only motion. When applied to a non-rigid body, it can cause both a change in size/shape and motion.

• Mathematical Definition of Force: Force is defined as the rate of change of linear momentum:     $F = \frac{dp}{dt}$     $F = \frac{d(mv)}{dt} = ma$     (Note: This assumes mass $m$ is constant).

• Force as a Vector: Force is a vector quantity with the S.I. unit of newton ($N$) or gravitational unit kilogram-force ($kgf$).

• Unit Conversion: $1\,kgf = g\,N$. Using the average acceleration due to gravity, $1\,kgf = 9.8\,N$.

Translational and Rotational Motions

• Linear or Translational Motion: This occurs when a force acts on a stationary rigid body that is free to move, causing it to move in a straight path in the direction of the force. Example: Pushing a ball lying on the floor.

• Rotational Motion: This occurs if 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 known as the turning effect of the force. Example: A wheel pivoted at its centre with a tangential force applied to the rim, or a door rotating about hinges.

Moment (Turning Effect) of a Force or Torque

• Conceptual Definition: If a body is pivoted at a point $O$ and a force $F$ is applied with a line of action horizontal to the pivot, it cannot produce linear motion because the body is fixed. Instead, it turns the body about the vertical axis passing through $O$.

• Measurement of Torque: The moment of a force (torque) is equal to the product of the magnitude of the force ($F$) and the perpendicular distance ($OP$) of the line of action of the force from the axis of rotation.     $\text{Moment of force} = F \times OP$

• Factors Affecting Turning Effect:

  1. The magnitude of the force applied.

  2. The 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 should be applied at a point where the perpendicular distance from the axis of rotation is at its maximum.

Units and Convention of Moments

• Units of Moment of Force:

  • S.I. Unit: newton $\times$ metre, abbreviated as $N\,m$. Note: $N\,m$ is not written as "joule" for torque because torque is a vector, whereas work/energy are scalars.

  • C.G.S. Unit: dyne $\times$ cm.

  • Gravitational S.I. Unit: $kgf\,m$.

  • Gravitational C.G.S. Unit: $gf\,cm$.

• Conversions:

  • $1\,N\,m = 10^5\,dyne \times 10^2\,cm = 10^7\,dyne\,cm$

  • $1\,kgf\,m = 9.8\,N\,m$

  • $1\,gf\,cm = 980\,dyne\,cm$

• Directional Convention:

  • Anticlockwise Moment: Taken as positive ($+$). It acts along the axis of rotation outwards.

  • Clockwise Moment: Taken as negative ($-$). It acts along the axis of rotation inwards.

• Determinants of Rotation Direction: The direction depends on the point of application of the force and the direction of the force itself. Examples include a disc pivoted at the centre or an axle.

Common Examples of Moment of Force

• Opening a Door: Handles are provided at the maximum distance from the hinges (the free end) so that a smaller force produces the required torque. Applying force at the hinge results in zero torque.

• Hand Flour Grinder: A handle is placed near the rim (maximum distance from the centre) to make rotation easier around the central iron pivot.

• Steering Wheel: A small tangential force is applied to the rim. The sense of rotation is changed by changing the point of application (e.g., applying force at the top vs. bottom).

• Bicycle Toothed Wheel: A large toothed wheel is used with the foot pedal to increase the perpendicular distance from the axle, requiring less force to turn the rear wheel via the chain.

• Spanner (Wrench): A long handle is used to produce a large moment of force with a small normal force. Clockwise rotation tightens the nut; anticlockwise loosens it.

Couple

• Definition: A single force applied to a pivoted body cannot cause rotation alone; rotation is produced by a pair of forces called a "couple." A couple consists of two equal and opposite parallel forces that do not act along the same line.

• Internal Reactions: In many examples, the couple consists of the external force and the reaction force at the pivot. The reaction force at the pivot is equal and opposite to the applied force, but its moment is zero because the distance is zero.

• Examples of Couples:

  • Turning a water tap.

  • Tightening the cap of an inkpot.

  • Turning a key in a lock.

  • Winding a clock with a key.

  • Driving a bicycle pedal (two feet applying opposite forces).

• Moment of Couple: Calculated as the product of either force and the perpendicular distance between the two forces (called the couple arm).     $\text{Moment of couple} = F \times d$

Equilibrium of Bodies

• Definition: When a number of forces acting on a body produce no change in its state of rest or of linear/rotational motion, the body is in equilibrium.

• Static Equilibrium: The body remains at rest under the influence of several forces.

  • Example: A body pulled by equal and opposite forces on a table.

  • Example: A book on a table (weight balanced by normal reaction).

  • Example: A beam balance in a horizontal position.

• Dynamic Equilibrium: The body remains in the same state of motion (translational or rotational) under multiple forces.

  • Example: A rain drop falling at constant velocity (weight balanced by buoyancy and air friction).

  • Example: An aeroplane moving at a constant height.

  • Example: A stone whirled in a circular path at uniform speed.

  • Example: Planets orbiting the sun or electrons orbiting a nucleus.

• Conditions for Equilibrium:

  1. The resultant of all forces acting on the body must be zero.

  2. The algebraic sum of moments of all forces about any point must be zero (Sum of anticlockwise moments = Sum of clockwise moments).

Principle of Moments

• Statement: In equilibrium, the sum of clockwise moments is equal to the sum of anticlockwise moments about the axis of rotation.

• Verification: Using a metre rule suspended at its centre ($O$). Slotted weights ($W_1, W_2$) are placed at distances ($l_1, l_2$) from the centre. Equilibrium is achieved when $W_1 \times l_1 = W_2 \times l_2$.

• Numerical Example (Handle Length): If a $150\,N$ force opens a nut with a $40\,cm$ handle (Torque = $60\,N\,m$), then a $50\,N$ force requires a handle of length $L = \frac{60}{50} = 1.2\,m$.

Centre of Gravity (C.G.)

• Definition: The C.G. is the point about which the algebraic sum of the moments of weights of all constituent particles of the body is zero. The entire weight of the body ($W$) acts at this point.     $W = w_1 + w_2 + w_3 + \dots$

• Characteristics:

  • Depends on the distribution of mass and the shape of the body.

  • Changes if the body is deformed (e.g., a straight wire bent into a circle).

  • The C.G. does not have to be within the material of the body (e.g., the centre of a ring or hollow sphere).

• C.G. Positions for Regular Objects:

  • Rod: Mid-point of the rod.

  • Circular Disc/Ring: Geometric centre.

  • Sphere (Solid/Hollow): 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 (centroid).

  • Parallelogram/Rectangle/Square: Intersection of diagonals.

• Experimental Determination: For an irregular lamina, C.G. is found using a plumb line from three different suspension points ($a, b, c$). The intersection of these lines is the C.G. ($G$).

Uniform Circular Motion

• Definition: A particle moving with constant speed in a circular path.

• Nature of Motion: While speed is uniform, the direction of motion changes at every point. This means velocity is variable/non-uniform, and the motion is inherently accelerated.

• Direction of Velocity: At any instant, the direction of motion is along the tangent drawn at that point of the circular path.

• Comparison with Linear Motion:

  • Uniform Linear Motion: Speed and velocity are constant; acceleration is zero.

  • Uniform Circular Motion: Speed is constant; velocity is variable; acceleration is non-zero.

Centripetal and Centrifugal Forces

• Centripetal Force: A force directed towards the centre of the circle required to keep a body in circular motion by continuously changing its direction.

  • Example: Electrostatic attraction for electrons.

  • Example: Gravitational force for planets/moons.

  • Example: Tension in a string for a whirled stone.

• Centrifugal Force: A fictitious or virtual force that acts on a body moving in a circular path, directed away from the centre.

  • Nature: It is not a real force and is not the reaction to centripetal force (as action/reaction must act on different bodies).

  • Perspective: It is assumed by an observer within the rotating frame of reference (e.g., a person on a merry-go-round) to explain why an object appears stationary despite the centripetal tension.

  • String Breakage Scenario: If the string of a whirled stone breaks, an observer on the ground sees the stone fly off tangentially (due to inertia), while an observer in the rotating frame sees it move radially outward.

Questions and Discussion

• Q: Why is it easier to turn a steering wheel of a larger diameter?

  • A: A larger diameter provides a larger perpendicular distance from the axis of rotation, allowing a smaller force to produce the required moment of force.

• Q: Is centrifugal force a real force?

  • A: No, it is a fictitious or virtual force used only to describe motion within a rotating frame of reference.

• Q: At what point is the C.G. of a triangular lamina?

  • A: At the point of intersection of its medians.

• Q: What happens to the direction of acceleration in uniform circular motion?

  • A: The direction of acceleration changes continuously as it is always directed towards the centre, though its magnitude remains constant.

• Q: Calculation involving a See-Saw: Children of mass $30\,kg$ and $50\,kg$ sit at $2\,m$ and $2.5\,m$ from the pivot. Where should a $74\,kg$ man sit?

  • Calculation:         $\text{Anticlockwise Moment} = (30 \times 2) + (50 \times 2.5) = 60 + 125 = 185\,kgf\,m$         $\text{Clockwise Moment} = 74 \times x$         $74x = 185 \rightarrow x = 2.5\,m$

RETRY

Syllabus Coverage and Introduction to Force

• Syllabus Scope (i): Includes the concept of turning forces, moment of a force, forces in equilibrium, and the centre of gravity (C.G.). It covers elementary introductions to translational and rotational motions, torque and its units (C.G.S. and S.I.), and common examples like a door, steering wheel, and bicycle pedal. It also addresses clockwise and anticlockwise moments, conditions for translational and rotational equilibrium, the principle of moments, and the centre of gravity for regular and irregular bodies.

• Syllabus Scope (ii): Focuses on uniform circular motion as an example of constant speed despite the presence of acceleration (force), and explains the difference between centrifugal and centripetal forces.

• Force on Rigid vs. Non-Rigid Bodies: A force applied to a rigid body causes only motion. When applied to a non-rigid body, it can cause both a change in size/shape and motion.

• Mathematical Definition of Force: Force is defined as the rate of change of linear momentum: $F = \frac{dp}{dt}$ $F = \frac{d(mv)}{dt} = ma$ (Note: This assumes mass $m$ is constant).

• Force as a Vector: Force is a vector quantity with the S.I. unit of newton ($N$) or gravitational unit kilogram-force ($kgf$).

• Unit Conversion: $1\,kgf = g\,N$. Using the average acceleration due to gravity, $1\,kgf = 9.8\,N$.

Translational and Rotational Motions

• Linear or Translational Motion: This occurs when a force acts on a stationary rigid body that is free to move, causing it to move in a straight path in the direction of the force. Example: Pushing a ball lying on the floor.

• Rotational Motion: This occurs if 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 known as the turning effect of the force. Example: A wheel pivoted at its centre with a tangential force applied to the rim, or a door rotating about hinges.

Moment (Turning Effect) of a Force or Torque

• Conceptual Definition: If a body is pivoted at a point $O$ and a force $F$ is applied with a line of action horizontal to the pivot, it cannot produce linear motion because the body is fixed. Instead, it turns the body about the vertical axis passing through $O$.

• Measurement of Torque: The moment of a force (torque) is equal to the product of the magnitude of the force ($F$) and the perpendicular distance ($OP$) of the line of action of the force from the axis of rotation. $\text{Moment of force} = F \times OP$

• Factors Affecting Turning Effect:

  1. The magnitude of the force applied.

  2. The 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 should be applied at a point where the perpendicular distance from the axis of rotation is at its maximum.

Units and Convention of Moments

• Units of Moment of Force:

  • S.I. Unit: newton $\times$ metre, abbreviated as $N\,m$. Note: $N\,m$ is not written as "joule" for torque because torque is a vector, whereas work/energy are scalars.

  • C.G.S. Unit: dyne $\times$ cm.

  • Gravitational S.I. Unit: $kgf\,m$.

  • Gravitational C.G.S. Unit: $gf\,cm$.

• Conversions:

  • $1\,N\,m = 10^5\,dyne \times 10^2\,cm = 10^7\,dyne\,cm$

  • $1\,kgf\,m = 9.8\,N\,m$

  • $1\,gf\,cm = 980\,dyne\,cm$

• Directional Convention:

  • Anticlockwise Moment: Taken as positive ($+$). It acts along the axis of rotation outwards.

  • Clockwise Moment: Taken as negative ($-$). It acts along the axis of rotation inwards.

• Determinants of Rotation Direction: The direction depends on the point of application of the force and the direction of the force itself. Examples include a disc pivoted at the centre or an axle.

Common Examples of Moment of Force

• Opening a Door: Handles are provided at the maximum distance from the hinges (the free end) so that a smaller force produces the required torque. Applying force at the hinge results in zero torque.

• Hand Flour Grinder: A handle is placed near the rim (maximum distance from the centre) to make rotation easier around the central iron pivot.

• Steering Wheel: A small tangential force is applied to the rim. The sense of rotation is changed by changing the point of application (e.g., applying force at the top vs. bottom).

• Bicycle Toothed Wheel: A large toothed wheel is used with the foot pedal to increase the perpendicular distance from the axle, requiring less force to turn the rear wheel via the chain.

• Spanner (Wrench): A long handle is used to produce a large moment of force with a small normal force. Clockwise rotation tightens the nut; anticlockwise loosens it.

Couple

• Definition: A single force applied to a pivoted body cannot cause rotation alone; rotation is produced by a pair of forces called a "couple." A couple consists of two equal and opposite parallel forces that do not act along the same line.

• Internal Reactions: In many examples, the couple consists of the external force and the reaction force at the pivot. The reaction force at the pivot is equal and opposite to the applied force, but its moment is zero because the distance is zero.

• Examples of Couples:

  • Turning a water tap.

  • Tightening the cap of an inkpot.

  • Turning a key in a lock.

  • Winding a clock with a key.

  • Driving a bicycle pedal (two feet applying opposite forces).

• Moment of Couple: Calculated as the product of either force and the perpendicular distance between the two forces (called the couple arm). $\text{Moment of couple} = F \times d$

Equilibrium of Bodies

• Definition: When a number of forces acting on a body produce no change in its state of rest or of linear/rotational motion, the body is in equilibrium.

• Static Equilibrium: The body remains at rest under the influence of several forces.

  • Example: A body pulled by equal and opposite forces on a table.

  • Example: A book on a table (weight balanced by normal reaction).

  • Example: A beam balance in a horizontal position.

• Dynamic Equilibrium: The body remains in the same state of motion (translational or rotational) under multiple forces.

  • Example: A rain drop falling at constant velocity (weight balanced by buoyancy and air friction).

  • Example: An aeroplane moving at a constant height.

  • Example: A stone whirled in a circular path at uniform speed.

  • Example: Planets orbiting the sun or electrons orbiting a nucleus.

• Conditions for Equilibrium:

  1. The resultant of all forces acting on the body must be zero.

  2. The algebraic sum of moments of all forces about any point must be zero (Sum of anticlockwise moments = Sum of clockwise moments).

Principle of Moments

• Statement: In equilibrium, the sum of clockwise moments is equal to the sum of anticlockwise moments about the axis of rotation.

• Verification: Using a metre rule suspended at its centre ($O$). Slotted weights ($W_1, W_2$) are placed at distances ($l_1, l_2$) from the centre. Equilibrium is achieved when $W_1 \times l_1 = W_2 \times l_2$.

• Numerical Example (Handle Length): If a $150\,N$ force opens a nut with a $40\,cm$ handle (Torque = $60\,N\,m$), then a $50\,N$ force requires a handle of length $L = \frac{60}{50} = 1.2\,m$.

Centre of Gravity (C.G.)

• Definition: The C.G. is the point about which the algebraic sum of the moments of weights of all constituent particles of the body is zero. The entire weight of the body ($W$) acts at this point. $W = w_1 + w_2 + w_3 + \dots$

• Characteristics:

  • Depends on the distribution of mass and the shape of the body.

  • Changes if the body is deformed (e.g., a straight wire bent into a circle).

  • The C.G. does not have to be within the material of the body (e.g., the centre of a ring or hollow sphere).

• C.G. Positions for Regular Objects:

  • Rod: Mid-point of the rod.

  • Circular Disc/Ring: Geometric centre.

  • Sphere (Solid/Hollow): 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 (centroid).

  • Parallelogram/Rectangle/Square: Intersection of diagonals.

• Experimental Determination: For an irregular lamina, C.G. is found using a plumb line from three different suspension points ($a, b, c$). The intersection of these lines is the C.G. ($G$).

Uniform Circular Motion

• Definition: A particle moving with constant speed in a circular path.

• Nature of Motion: While speed is uniform, the direction of motion changes at every point. This means velocity is variable/non-uniform, and the motion is inherently accelerated.

• Direction of Velocity: At any instant, the direction of motion is along the tangent drawn at that point of the circular path.

• Comparison with Linear Motion:

  • Uniform Linear Motion: Speed and velocity are constant; acceleration is zero.

  • Uniform Circular Motion: Speed is constant; velocity is variable; acceleration is non-zero.

Centripetal and Centrifugal Forces

• Centripetal Force: A force directed towards the centre of the circle required to keep a body in circular motion by continuously changing its direction.

  • Example: Electrostatic attraction for electrons.

  • Example: Gravitational force for planets/moons.

  • Example: Tension in a string for a whirled stone.

• Centrifugal Force: A fictitious or virtual force that acts on a body moving in a circular path, directed away from the centre.

  • Nature: It is not a real force and is not the reaction to centripetal force (as action/reaction must act on different bodies).

  • Perspective: It is assumed by an observer within the rotating frame of reference (e.g., a person on a merry-go-round) to explain why an object appears stationary despite the centripetal tension.

  • String Breakage Scenario: If the string of a whirled stone breaks, an observer on the ground sees the stone fly off tangentially (due to inertia), while an observer in the rotating frame sees it move radially outward.

Questions and Discussion

• Q: Why is it easier to turn a steering wheel of a larger diameter?

  • A: A larger diameter provides a larger perpendicular distance from the axis of rotation, allowing a smaller force to produce the required moment of force.

• Q: Is centrifugal force a real force?

  • A: No, it is a fictitious or virtual force used only to describe motion within a rotating frame of reference.

• Q: At what point is the C.G. of a triangular lamina?

  • A: At the point of intersection of its medians.

• Q: What happens to the direction of acceleration in uniform circular motion?

  • A: The direction of acceleration changes continuously as it is always directed towards the centre, though its magnitude remains constant.

• Q: Calculation involving a See-Saw: Children of mass $30\,kg$ and $50\,kg$ sit at $2\,m$ and $2.5\,m$ from the pivot. Where should a $74\,kg$ man sit?

  • Calculation: $\text{Anticlockwise Moment} = (30 \times 2) + (50 \times 2.5) = 60 + 125 = 185\,kgf\,m$ $\text{Clockwise Moment} = 74 \times x$ $74x = 185 \rightarrow x = 2.5\,m$