The Turning Effect of Forces: Comprehensive Study Notes

Introduction to the Turning Effect of Forces

• Conceptual Overview: A force can exert a turning effect, allowing an object to rotate around a fixed point known as a pivot or fulcrum.

• Balance: When the anticlockwise turning effects of forces are exactly equalized by the clockwise turning forces, the object remains in balance and does not rotate.

• Real-World Applications:

  • See-saws: Children of different weights can balance by adjusting their distance from the pivot.

  • Cranes: Large concrete counterbalance weights are used to offset the load and the long load arm to prevent the support tower from bending or collapsing.

  • Human Anatomy: Muscles exert forces that cause body parts to turn around joints like elbows and knees.

  • Everyday Tasks: Turning door handles, opening doors, or using a knife to pry the lid off a paint tin.

• Dynamics: Unbalanced forces cause objects to accelerate or decelerate, while forces acting through pivots (or creating moments) cause rotational motion.

Opening a Door: The Advantage of Leverage

• The Experiment: In a trial to hold a door closed against a person opening it:

  • If the person holding the door pushes at a distance no further than $20\,cm$ from the hinge.

  • If the person opening the door pulls on the handle (typically much further from the hinge).

• Outcome: The person at the handle opens the door easily because of greater leverage.

• Key Insight: The turning effect depends on two variables: the magnitude of the force applied and the distance from the pivot (hinge) at which that force is applied.

The Moment of a Force

• Definition: The turning effect of a force about a hinge or pivot is called its moment.

• Mathematical Formula:

  • $\text{moment of a force (Nm)} = \text{force, F (N)} \times \text{perpendicular distance from pivot, d (m)}$

  • $\text{moment} = F \times d$

• Units: Measured in newton metres ($Nm$).

• Precision in Measurement:

  • Line of Action: To achieve the maximum turning effect, the force should be applied perpendicular ($90^{\circ}$) to the lever.

  • Zero Moment: If the line of action of the force passes directly through the pivot point, it has no turning effect.

  • Perpendicular Distance: The distance used in the calculation must be the perpendicular distance from the line of action of the force to the pivot.

Principle of Moments and Balance

• The Condition for Equilibrium: An object is in balance (no net rotation) when:

  • $\sum \text{anticlockwise moments} = \sum \text{clockwise moments}$

• See-saw Example (Balanced vs. Unbalanced):

  • Case A: A $200\,N$ child at $1.5\,m$ (anticlockwise moment = $200\,N \times 1.5\,m = 300\,Nm$) and a $300\,N$ child at $1.5\,m$ (clockwise moment = $300\,N \times 1.5\,m = 450\,Nm$). The see-saw is unbalanced and rotates clockwise.

  • Case B: The $200\,N$ child at $1.5\,m$ ($300\,Nm$ anticlockwise) and the $300\,N$ child moves to $1.0\,m$ from the pivot (clockwise moment = $300\,N \times 1.0\,m = 300\,Nm$). The see-saw is now balanced.

• Example 1: Door Calculation:

  • Person A pushes with $200\,N$ at $20\,cm$ from the hinge.

  • Person B pulls the handle at $80\,cm$ from the hinge.

  • To balance: $F \times 80\,cm = 200\,N \times 20\,cm \Rightarrow 80F = 4000 \Rightarrow F = 50\,N$.

  • To actually open the door, Person B must apply a force F > 50\,N.

Principles of Crane Operation

• Components: Load arm, crosspiece, controller's cabin, mast, and counterbalance weights.

• Mechanism: The load arm is long to reach across sites and move loads. To balance the moment created by the load on the long arm, large concrete blocks (counterbalance weights) are used on the short arm.

• Placement: Counterbalances must be very heavy because they are positioned closer to the pivot (where the tower supports the crosspiece) than the load usually is.

Centre of Gravity

• Definition: The centre of gravity (or centre of mass) is the point where the entire weight of an object appears to act.

• Ruler Balance Experiment: A uniform ruler balances at its exact center point because the mass is spread equally. Supporting it at this point results in no net turning moment.

• Extension: Regular Shapes: For uniform materials, the centre of gravity is located where the axes of symmetry intersect:

  • Rectangle: Intersection of diagonals.

  • Square: Intersection of diagonals.

  • Equilateral Triangle: Point where the three axes of symmetry meet.

  • Symmetry Test: Place a plane mirror along an axis; if the reflection matches the original shape, it is an axis of symmetry.

Objects Not Pivoted at the Centre of Gravity

• Turning Effect of Weight: If an object is not supported at its centre of gravity, its own weight creates a turning effect about the pivot.

• Example 2: Ruler with Offset Pivot:

  • A uniform metre ruler ($100\,cm$ total length) has a mass of $0.12\,kg$.

  • Centre of gravity is at the $50\,cm$ mark.

  • Pivot is placed at the $40\,cm$ mark.

  • A brass weight (mass $m$) is placed at the $20\,cm$ mark to balance it.

  • Distances from Pivot:

    • Brass weight to pivot = $40\,cm - 20\,cm = 20\,cm$.

    • Ruler’s weight (at $50\,cm$) to pivot = $50\,cm - 40\,cm = 10\,cm$.

  • Balance Equation:

    • $(m \times g) \times 20\,cm = (0.12\,kg \times g) \times 10\,cm$

    • $20m = 1.2 \Rightarrow m = 0.06\,kg$.

Stability and Center of Gravity

• Definition of Stability: A stable object is difficult to knock over and returns to its original position when released after a push.

• Variables Affecting Stability:

  • Height of Centre of Gravity: A lower centre of gravity increases stability.

  • Width of Base: A broader base increases stability.

• Examples:

  • Ship's Decanter: Features a very wide base and low centre of gravity to prevent tipping on a moving ship.

  • Unstable Objects: Have high centres of gravity and narrow bases; only a small displacement is needed for the line of action of the weight to fall outside the base, causing the object to topple.

Forces on a Beam

• Statics: When a boy stands on a beam (like a bridge over a stream), the total upward forces from the supports must equal the total downward force (the weight of the boy and the beam).

• Distribution of Force:

  • Middle Position: If the weight is exactly in the center, the upward forces at both supports are equal (e.g., a $400\,N$ boy results in $200\,N$ upward force at each end).

  • End Position: If the boy stands directly over one support, that support carries the entire $400\,N$ weight, while the other carries $0\,N$.

  • Proportional Distribution: If the boy stands one-quarter of the way along the beam ($0.25L$ from one end), the nearest support provides three-quarters of the upward force ($300\,N$), and the further support provides one-quarter ($100\,N$).

Chapter Questions and Skills Analysis

• Question 1 (Identifying Moments):

  • A: $10\,N$ applied at a distance from the pivot.

  • B: $10\,N$ applied directly at the pivot (moment = $0$).

  • C: Two $5\,N$ forces acting in opposite directions at equal distances (creates a couple/summed moment).

• Question 2 (Balanced See-saws):

  • A: $150\,N$ at $2.5\,m$ vs. $150\,N$ at $2.5\,m$ (Balanced).

  • B: $150\,N$ at $2.5\,m$ ($375\,Nm$) vs. $200\,N$ at $2\,m$ ($400\,Nm$) (Clockwise rotation).

  • C: $120\,N$ at $2.5\,m$ ($300\,Nm$) vs. $300\,N$ at $1\,m$ ($300\,Nm$) (Balanced).

  • D: $150\,N$ at $2.5\,m$ ($375\,Nm$) vs. $175\,N$ at $2\,m$ ($350\,Nm$) (Anticlockwise rotation).

• Question 3 (Bookshelf Calculations):

  • Beam length = $2\,m$, supports at P and Q.

  • Middle Book ($10\,N$): P = $5\,N$, Q = $5\,N$.

  • Book $50\,cm$ from Q: Distance from P = $1.5\,m$. Taking moments about P: $10\,N \times 1.5\,m = F_Q \times 2\,m \Rightarrow F_Q = 7.5\,N$. Then $F_P = 10\,N - 7.5\,N = 2.5\,N$.