Stability and Equilibrium Study Notes

Stability and Equilibrium

Year 12 - Translational Equilibrium

Today's Learning Outcomes

By the end of this lesson, you will be able to:

  • Explain the concepts of centre of mass and stability

  • Distinguish between stable, unstable, and neutral equilibrium

  • Apply the concept of translational equilibrium (ΣF = 0)

  • Solve equilibrium problems using vector components

Interactive Activity

In pairs, take turns to:

  1. Stand tall while someone tries to push you over

  2. Spread legs in a squat (like a sumo wrestler) while someone tries to push you over

Observations and Discussions

  • What did you notice?

  • Which position was more stable?

  • Why was it harder to push someone over in the squat position?

  • What physics concepts explain this?

Basic Physics Concepts

To understand, analyze and solve problems involving stability and equilibrium, we need to understand some basic concepts such as:

  • Equilibrium of forces

  • Centre of mass (also called Centre of gravity)

  • Stability

Equilibrium of Forces

  • Newton’s First Law: An object at rest remains at rest and an object in motion remains in constant motion unless an external force acts on it.

Centre of Mass and Stability

  • Centre of Mass Definition:

    • The center of mass of a system of particles is the point that moves as though all of the system's mass were concentrated at that point, and all external forces were applied there.

  • Key Points about Centre of Mass:

    • The mass of a body is distributed throughout the body. However, for practical purposes, we can consider the mass to act through a single reference point called the "centre of mass".

    • The center of mass can be referred to interchangeably as the centre of gravity (assuming a uniform gravitational field).

    • This point can be outside the object itself (e.g., in cases of a hollow ring or a high jumper).

    • The position can change with body configuration, such as with arms and legs changing position.

    • It is the point of balance for the body.

Stability Principles

  • Key Principle of Stability:

    • A body will NOT overturn (topple) as long as a vertical line through the centre of gravity passes through its base of support.

  • To Minimize Toppling:

    1. Lower the centre of gravity.

    2. Widen the base of support.

    • This explains why the sumo squat position was more stable compared to standing tall.

Types of Equilibrium

  1. Stable Equilibrium

    • When displaced, the object returns to its original position.

    • Centre of mass RISES, and potential energy (PE) increases.

    • Example: A bowl resting on a table or a pendulum at rest.

  2. Unstable Equilibrium

    • When displaced, the object moves further away from its original position.

    • Centre of mass LOWERS, and potential energy (PE) decreases.

    • Example: A pencil standing on its tip or an inverted pendulum.

  3. Neutral Equilibrium

    • When displaced, the object stays in its new position.

    • Centre of mass remains at the same height, and potential energy (PE) remains constant.

    • Example: A ball on a flat surface or a wheel.

Translational Equilibrium

  • Definition: If a system is in translational equilibrium, then it is NOT accelerating in any direction, and the net force is zero.

  • Mathematical Representation:

    • extSFextnet=0ext{SF}_{ ext{net}} = 0

    • This implies that the sum of forces in all directions is zero:

    • extSF<em>exthorizontal=0ext{SF}<em>{ ext{horizontal}} = 0 (or extSF</em>extleft−right=0ext{SF}</em>{ ext{left-right}} = 0)

    • extSF<em>extvertical=0ext{SF}<em>{ ext{vertical}} = 0 (or extSF</em>extup−down=0ext{SF}</em>{ ext{up-down}} = 0)

Worked Examples

Example 2.2.1 (One Dimension)

  • Situation: Three students are standing on a plank bridging a stream.

  • Plank Mass: 25.0 kg

  • Students' Masses: 40.0 kg, 30.0 kg, and 35.0 kg.

  • Force from Left Bank on Plank: 725 N (upward)

  • Calculate: Force of the right bank on the plank, assuming translational equilibrium.

  • Use: g=9.80extNkg−1g = 9.80 ext{ N kg}^{-1}.

Example Try Yourself 2.2.1

  • Situation: Three cars parked on a beam bridge with mass 525 kg.

  • Force from Pillar X: 2.00imes104extN2.00 imes 10^{4} ext{ N} (upward)

  • Calculate: Force provided by pillar Y, assuming translational equilibrium.

  • Use: g=9.80extNkg−1g = 9.80 ext{ N kg}^{-1}.

Example 2.2.2 (Two Dimensions)

  • Situation: An advertising sign has a mass of 45.0 kg, hung by two cables at an angle of 30.0° to the vertical.

  • Calculate: Tension in each cable when the sign is suspended.

  • Use: g=9.80extNkg−1g = 9.80 ext{ N kg}^{-1}.

Example Try Yourself 2.2.2

  • Situation: A concrete beam of mass 1510 kg is lifted by cables at angles of 60.0°.

  • Calculate: Tensions in cable 1 and cable 2, with constant upward velocity of 2.00 m/s.

  • Use: g=9.80extNkg−1g = 9.80 ext{ N kg}^{-1}.

Translational Equilibrium Examples

Example 1

  • Situation: Two people on a see-saw; one weighs 25.0 kg and the other 55.0 kg.

  • Task: Determine the reaction force at the fulcrum if the see-saw is in translational equilibrium.

Example 2

  • Situation: A metal sphere of mass 200.0 kg is pulled horizontally until the cable makes a 15.0° angle with the vertical.

  • Task: Find the horizontal force required and the tension in the cable.

Exercises

Exercise 1

  • Situation: A tightrope walker with a mass of 76.0 kg stands in the middle of a tight rope that makes an angle of 10.0° with the horizontal.

  • Task: Determine the tension in the ropes.

Exercise 2

  • Situation: A sign with mass 50.0 kg is supported by cables.

  • Calculate: Tension in the cables; one cable is horizontal, the other at an angle of 25.0° to the horizontal.

Exercise 3

  • Situation: Traffic lights weighing 800.0 N are suspended by two cables with angles of 50.0° and 40.0° to the horizontal.

  • Task: Find the tension in each cable.

Review Questions

Question 1

  • Three forces acting on an object:

    • F1 = 2.00 N (west),

    • F2 = 3.00 N (south),

    • F3 = 6.00 N (east).

  • Tasks:
    a) Determine the resultant force on the object.
    b) Identify the additional necessary force to achieve equilibrium.

Question 2

  • A pot-plant hangs from a wall using a rod and rope. The pot-plant exerts a vertical force of 200 N.

  • Calculate: Tension in the rope if the rod exerts a horizontal force away from the wall.

Question 3

  • A crate of mass 50.0 kg is on a ramp inclined at 40°.

  • Tasks:
    a) Determine the minimum force required to stop it from moving down.
    b) Calculate the reaction force exerted by the ramp.
    c) Find the minimum horizontal force needed to keep it stationary on the ramp.

Question 4

  • Altering the angle of a cable to 25.0°, find the force in the cable and in the jib.

Question 5

  • A bag of sand weighing 100 N is pulled aside so the rope makes a 25.0° angle with the vertical.

  • Find: Horizontal force and tensile force in the rope.

2023 WATP Sem 1 (Section 1)

Question 8

(5 marks)

  • A water drum hangs in a shed using two ropes: rope 1 at a right angle with the wall and rope 2 makes angle 0 with the roof.
    Tasks:
    a) Which rope experiences the greatest tensile force?
    b) Explain your answer using principles of equilibrium and relevant formulas.

Summary

  • Translational equilibrium in one dimension is represented mathematically as: extΣF=0ext{ΣF} = 0.

  • In two dimensions: extΣF<em>x=0ext{ΣF}<em>{x} = 0 and extΣF</em>y=0ext{ΣF}</em>{y} = 0.

  • Equilibrium of forces can be visually represented in a closed vector diagram.

  • Calculate x and y components to identify forces in equilibrium.

Key Questions

  1. When does translational equilibrium occur?

    • A. A net force acts through the centre of mass causing no acceleration.

    • B. A net force acts through the centre of mass causing an acceleration.

    • C. No net force acts through the centre of mass resulting in acceleration.

    • D. No net force acts through the centre of mass ensuring no acceleration.

  2. An object in translational equilibrium:

    • A. Must be stationary.

    • B. Must move at constant velocity.

    • C. Must experience translational acceleration.

    • D. Must not experience translational acceleration.

  3. Calculate tension in the string supporting a birdfeeder mass of 355 g in translational equilibrium.

  4. Calculate mass of a pendulum bob hanging when tension in chain is 7.50 N.

  5. Calculate tension forces in each wire cable supporting a kitchen rail along with tools and frying pan.

  6. Calculate tension in one of four cables supporting two window cleaners and a platform.

  7. Draw force vector from a child pushing a shopping trolley for translational equilibrium.

  8. Calculate tensions in cables supporting a 112 kg bowling ball.

  9. What is the maximum mass of a picture that a wire can support when maximum force is 40.0 N?

  10. Calculate tension in a rope tied between two posts when the performer stands centrally making a 10.0° angle with horizontal.