4.2 Moments

Figure 1 shows a simple toy mobile. This is a suspended ornament that rotates and swings in a slight breeze. A star, moon and planet toy are suspended by cotton suspensions from the ceiling. 

Figure 1

The weights of the objects are such that they are in equilibrium, with the weight of the planet as 1.35 N. The weights of the horizontal bars are negligible. 

Calculate the weight of the star.

A child pulls vertically downward on the moon toy. 

Explain the effect this would have on the equilibrium of the system.

A different mobile is set up in Figure 2 which is also in equilibrium. Added objects are a comet, astronaut and rocket. 

Figure 2

The weight of the rocket is 1.50 N. The weights of the horizontal bars are still negligible. 

Calculate the distance between the suspension hanging the moon and planet to the suspension hanging the first bar from the ceiling.

A bench consists of a plank which is resting in a horizontal position on two supports B and C. The plank is modelled as a uniform rod AD of length 2.7 m and mass 15 kg. The supports at B and C are 0.5 m from each end of the plank, as shown in Figure 1. 

Figure 1

Two students, Rose and Quinn, sit on the plank. Rose has mass 55 kg and sits in the middle of the plank and Quinn has a mass 62 kg and sits at the end A. The plank remains horizontal and in equilibrium. The students can be modelled as point masses. 

 Calculate: 

   (i)    The magnitude of the normal reaction force at B 

   (ii    The magnitude of the normal reaction force at C

Whilst Quinn stays at point A, Rose now moves along the plank to a point X. 

Given that the plank remains horizontal and in equilibrium, and that the magnitude of the normal reaction force at B is now twice the magnitude of the normal reaction force at C, calculate the distance BX.

A pole PR has a length 4 m and a weight W newton. The pole is held in a horizontal position in equilibrium by two vertical ropes attached to the pole at the points P and R where PR = 4 m, as shown in Figure 2. 

Figure 2

A load of weight 12 N is attached to the rod at R. The pole is uniform and the ropes are light inextensible strings. 

Given that the tension in the rope attached to the pole at Q is seven times the tension in the rope attached to the pole at P, calculate the value of W.

Q is now moved to a point further along the pole until the length QR is 0.8 m , as shown in Figure 3. The pole still remains in equilibrium. The load of 12 N is now moved to a point x metres from P. The beam remains in equilibrium in a horizontal position. 

Figure 3

The rope at Q will break if its tension exceeds 154 N. The rope at P cannot break. 

Find the range of possible positions on the beam where the load can be attached without the rope at Q breaking.

Figure 1 shows a uniform ladder resting against a vertical wall. The ladder is at an angle of θ to the ground. 

Figure 1

The resultant force from the wall on the ladder is twice the weight of the ladder. 

Show that the angle θ is around 14º. 

You may use the trigonometric identity: tan(θ) = .

Suggest the changes to the forces acting on the ladder that occur when someone climbs it.

A uniform rod AB of mass m and length 2a is freely hinged to a fixed point A. An object of mass m is attached at B and the rod is held in equilibrium at an angle θ to the horizontal by a force of magnitude F acting at point C on the rod. The force acts at right angles to AB where AC = b as shown in Figure 2. 

Figure 2

In terms of a, b, m and θ, determine the equations for: 

            (i) The horizontal component of the force acting on the rod at A 

            (ii)The vertical component of the force acting on the rod at A

Given that mass m is 2 kg, a is 4 m and b is 1.3 m and the angle θ is 70º, show that the resultant force acting on the rod at A is around 62 N.

A metre ruler has a support pivot 0.20 m from one end. The centre of mass of the ruler lies 0.43 m from this end. A1.5 kg mass is positioned 0.30 m from the opposite end of the ruler to the pivot. An upwards force of 30 N is applied at this end of the ruler at an angle of 20º. 

Calculate the mass of the ruler.

A shaduf is a device used to lift water from a well. It consists of an upright support to which a uniform beam is pivoted. It can be assumed that the weight of the beam is negligible. On one end of the beam is a counterweight, and on the other a bucket which can hold water, as shown in Figure 1. 

Figure 1

The counterweight is of uniform material and has a mass of 7.1 kg and is 0.40 m long. The bucket has a weight of 165 N and a capacity of 0.26 m³. When the bucket is half full, a force is required at the end of the beam to lift the bucket and water. 

Calculate the value of this force when the beam is horizontal. 

            Density of water = 1 g cm^-3

A student says that if the weight of the beam is not considered to be negligible, then the force at the end of the beam would become larger to counteract this weight and keep the beam in equilibrium. 

Discuss whether the student is correct.

A waiter holds a uniform tray of mass 150 g horizontally in one hand between their fingers and thumb as shown in Figure 1. 

Figure 1

X, Y and W are the three forces acting on the tray. 

Describe the relationships between the forces that must be satisfied if the tray is to remain horizontal and in equilibrium.

Calculate the length of the tray. Explain any assumptions made. 

Calculate the magnitude of the force X and Y.

The waiter places a glass of water on the tray.

State and explain where the glass should be position on the tray if the force X is to have the same value as in part (c).

It is said that Archimedes used huge levers to sink Roman ships invading the city of Syracuse. A possible system is shown in the figure 1 where a rope is hooked on to the front of the ship and the lever is pulled by several men. 

Figure 1

Calculate the mass of the ship if its weight is 4.6 × 10⁴ N.

Calculate the moment of the ship’s weight about point P. State an appropriate unit for your answer.

Calculate the minimum vertical force, T, required to start to raise the front of the ship. Assume the ship pivots about point P.

Calculate the minimum force, F, that must be exerted to start to raise the front of the ship.

The torque of a couple is given by 

                  torque = F × s 

With the aid of a diagram explain what is meant by a couple. Label Fand s on your diagram. State the appropriate unit for the torque of a couple.

The see-saw shown in Figure 1 consists of a uniform beam freely pivoted at the centre of the beam. Two children sit opposite each other so that the see-saw is in equilibrium.

Figure 1

Explain why the see-saw is in equilibrium.

Explain why the weight of the beam does not affect equilibrium.

Figure 2 shows the see-saw with three children of weights 460 N, 310 N and 240 N sitting so that the see-saw is in equilibrium. 

Figure 2

Calculate the distance, d.