1. (a) For the mechanism shown in FIGURE 1 determine for the angle θ = 45°:

(i) the velocity of the piston relative to the fixed point O(V_BO)

(ii) the angular velocity of AB about point A (i.e. ω_AB)

(iii) the acceleration of point B relative to A (a_BA).

Note: Link AB is horizontal when θ = 45°

(b) Determine the value of the angle θ (measured from vertical) when:

(i) the velocity of point B = 0

(ii) the angular velocity of link AB a maximum.

(c) What is the maximum angular velocity of link AB?

2. (a) A shaft 2 m long rotates at 1500 revs min^-1 between bearings as shown in FIGURE 2. The bearings experience forces of 5 kN and 3 kN acting in the same plane as shown. A single mass is to be used to balance the shaft, so that the reactions are zero. The mass is to be placed at a radius of 200 mm from the shaft centre, 180° from the direction of the bearing reactions. Determine the size and position (a and b) of the mass to be used.

(b) The shaft in part (a) is to be balanced using two masses (m₁ and m₂) placed 0.5 m and 1.5 m from end A and 180° from the direction of the bearing reactions, each on radius arms 100 mm long. Calculate the sizes of m₁ and m₂.

3. A gearbox and flywheel are as shown in FIGURE 4. The output shaft rotates in the opposite direction to the input shaft at 5 times its speed. The gearbox has an efficiency of 92%.

If the flywheel is solid, has a mass of 50 kg, a diameter of 1.5 m and is to accelerate from rest to 300 revs min^-1 in 1 minute:

(a) Calculate the torque required at input T₁.

(b) Calculate the magnitude and direction of the torque required to hold the gearbox stationary (holding torque Th). Show the direction of the holding torque applied to the shaft with the aid of a sketch.

(c) Plot a graph of the input power against time when taking the flywheel from rest to 300 revs min^-1.