Four pulleys are connected by two belts,

Question

pulley A i (radius 17 cm) is the drive pulley and it rotates at 11 rad/s. pulley B (radius 11 cm) is connected by belt 1 to pulley A. Pulley B' (radius 5.1 cm) is concentric with pulley B and is rigidly attached to it. Pulley C (radius 22 cm) is connected by belt 2 to pulley B'.

Calculate (a) The linear speed of a point on belt 1 (b) the angular speed of pulley B (c) the angular speed of pulley B' (d) the linear speed of a point on belt 2, and (e) the angular speed of pulley C

Answer 1

Pulley A (radius = r_A = 17cm) is the drive pulley, and rotates at angular velocity ω_A = 11 rad/s.

The linear speed of a point on its circumference is given as v_A = r_A ω_A .

The question asks the linear speed of a point on the belt 1 connecting pulleys A and B.

Assuming that the belt is not slipping on the circumference of pulley A, its linear speed is the same as the speed of a point on the circumference of A, that is : v₁ = r_A ω_A . = 17 × 11 cm/s

Note: remember; rads are not really dimensions; they are just ratios of <arc length subtended by an angle on a circle of a given radius> and the <radius>, so rad × cm is just cm)

Next, let us turn our attention to pulley B (radius = r_B = 11 cm).

If the belt isn't slipping on B, the linear speed of the belt 1 = linear speed of a point on circumference of B.

v₁ = v_B = r_B ω_B . ⇒ ω_B = v₁ / r_B .

Pulley B' is concentric with B and rigidly attached to its axle.

What this means is that the pulley B' is constrained to rotate at the same rate as the pulley B.

So, we may write ω_B = ω_B' .

In short, when two pulleys are connected by a belt, the relation between their motions is that the linear speeds of their circumferences is the same; whereas when two pulleys share the same axle, their angular velocities are the same.

Hope this helps.