four pulleys are connected by two belts. Pulley A (radius 15cm) is the drive pulley and it rotates at 10 rad/s. Pulley B (radius 10 cm) is connected by belt 1 to pulley A. Pulley B' (radius 5cm) is concentric with pulley B and is rigidly attached to it. Pulley C (radius 25cm) is connected by belt 2 to pulley B'. Calculate (a) the linear speed of 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 (hint: if the belt between two pulleys does not slip, the linear speeds at the rims of the two pulleys must be equal)

Question

four pulleys are connected by two belts. Pulley A (radius 15cm) is the drive pulley and it rotates at 10 rad/s. Pulley B (radius 10 cm) is connected by belt 1 to pulley A. Pulley B' (radius 5cm) is concentric with pulley B and is rigidly attached to it. Pulley C (radius 25cm) is connected by belt 2 to pulley B'. Calculate (a) the linear speed of 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 (hint: if the belt between two pulleys does not slip, the linear speeds at the rims of the two pulleys must be equal)

Draw the diagram, then work from A (given w) to linear speed, then the next pulley with the same linear speed, then that new jpulley has angular speed = linearspeed/radius. And so on.

Rigidly attached means the same angular speed.

Answer 1

To solve this problem, we will work step by step using the information given and the principles of pulley systems.

Step 1: Draw the diagram.
Draw a diagram illustrating the four pulleys and the two belts connecting them. Label the radii of the pulleys as given in the problem.

Step 2: Calculate the linear speed of a point on Belt 1 (a).
The linear speed of a point on Belt 1 can be calculated using the formula:
Linear speed = Angular speed * Radius.

Given that the drive pulley, Pulley A, rotates at 10 rad/s and has a radius of 15 cm, we can use the formula to calculate the linear speed of Belt 1:
Linear speed of Belt 1 = 10 rad/s * 15 cm = 150 cm/s.

So, the linear speed of a point on Belt 1 is 150 cm/s.

Step 3: Calculate the angular speed of Pulley B (b).
Since Belt 1 connects Pulley A to Pulley B, the linear speed of Belt 1 is the same as the linear speed of a point on Pulley B's rim.
Using the formula from Step 2, we can calculate the angular speed of Pulley B:
Angular speed of Pulley B = Linear speed of Belt 1 / Radius of Pulley B
= 150 cm/s / 10 cm = 15 rad/s.

So, the angular speed of Pulley B is 15 rad/s.

Step 4: Calculate the angular speed of Pulley B' (c).
Pulley B' is rigidly attached to Pulley B, which means it has the same angular speed.
Therefore, Angular speed of Pulley B' = Angular speed of Pulley B = 15 rad/s.

So, the angular speed of Pulley B' is also 15 rad/s.

Step 5: Calculate the linear speed of a point on Belt 2 (d).
The linear speed of a point on Belt 2 can be calculated using the formula from Step 2:
Linear speed of Belt 2 = Angular speed of Pulley B' * Radius of Pulley B'.
= 15 rad/s * 5 cm = 75 cm/s.

So, the linear speed of a point on Belt 2 is 75 cm/s.

Step 6: Calculate the angular speed of Pulley C (e).
Since Belt 2 connects Pulley B' to Pulley C, the linear speed on Belt 2 is the same as the linear speed of a point on Pulley C's rim.
Using the formula from Step 2, we can calculate the angular speed of Pulley C:
Angular speed of Pulley C = Linear speed of Belt 2 / Radius of Pulley C
= 75 cm/s / 25 cm = 3 rad/s.

So, the angular speed of Pulley C is 3 rad/s.

To recap:
(a) The linear speed of a point on Belt 1 is 150 cm/s.
(b) The angular speed of Pulley B is 15 rad/s.
(c) The angular speed of Pulley B' is 15 rad/s.
(d) The linear speed of a point on Belt 2 is 75 cm/s.
(e) The angular speed of Pulley C is 3 rad/s.

Remember, when solving problems like this, always start with the given information, apply the relevant formulas, and work step by step through the system.