Wheels connected by belts – problems and solutions

Wheels connected by belts – problems and solutions

1. Three wheels are connected as shown in the figure below.

If R_A = 10 cm, R_B = 4 cm, and R_C = 40 cm, then the ratio of the angular velocity of wheel A and wheel C is …

Known :

Radius of wheel A (r_A) = 10 cm

Radius of wheel B (r_B) = 4 cm

Radius of wheel C (r_C) = 40 cm

Wanted: the ratio of the angular velocity of wheel A and wheel C

Solution :

The angular velocity of wheel A and C

The circumference of wheel A is much larger than the circumference of wheel C. When the C wheel has been circularly rotated one circle (360^o), during the same time interval the wheel A not yet rotates one circle (360^o). Thus, the angular speed of the wheel A is not equal to the angular speed of the wheel C.

However, wheel A and wheel C are interconnected through ropes, so that in the same time interval, the distance traveled by the edge of the wheel A is equal to the distance traveled by the edge of the wheel C. Thus the linear speed of the edge of the wheel C (v_C) equal to the linear speed of the edge of the wheel A (v_A).

v_A = v_C

r_A ω_A = r_C ω_C

10 ω_A = 40 ω_C

ω_A / ω_C = 40 / 10

ω_A / ω_C = 4 / 1

2. Wheels B and C have the same axis of rotation and wheel A is tangent to wheel B. If radius of wheel A = radius of wheel C = 30 cm, the radius of wheel B = 60 cm, then determine the ratio of the linear speed between wheel A, B, and C.

Known :

Radius of wheel A (r_A) = 30 cm = 0.3 meters

Radius of wheel B (r_B) = 60 cm = 0.6 meters

Radius of wheel C (r_C) = 30 cm = 0.3 meters

Wanted : ratio of the linear speed between wheel A, B , and C.

Solution :

The linear speed of the edge of the wheel A :

Wheel A and wheel B are interconnected as shown in figure below, therefore the angular velocity of the wheel A is not equal to the angular velocity of the wheel B. This is because the circumference of wheel B is larger than wheel A. During the same time interval, when wheel A around one circle (360^o), wheel B not yet around one circle (360^o). However, during the same time interval, the distance traveled by the edge of wheel A is equal to the distance traveled by the edge of wheel B. Thus the linear velocity of the edge of the wheel A (v_A) is equal to the linear velocity of the edge of the wheel B (v_B).

The linear speed of the edge of wheel A :

v_A = r_A ω_A = 0.3 ω_A

The linear speed of the edge of the wheel B :

Wheel B and wheel B stick together, therefore, wheel B and wheel C rotate together. When wheel B around one circle (360^o) than during the same time interval, wheel C also around one circle (360^o). Since it rotates together, then the angular speed of wheel B (ω_B) is equal to the angular speed of wheel C (ω_C) = ω. But the linear speed of wheel B (vB) is not equal to the linear speed of wheel C (v_C)

The linear speed of the edge of wheel B :

v_B = r_B ω_B = 0.6 ω_B = 0.6 ω

The linear speed of the edge of wheel C :

v_C = r_C ω_C = 0.3 ω_C = 0.3 ω

The linear speed of the edge of wheel A (v_A) same as the linear speed of the edge of wheel B (v_B)

v_A = v_B

0.3 ω_A = 0.6 ω

ω_A = 0.6 ω / 0.3

ω_A = 2 ω

The linear speed of the edge of wheel A (v_A) :

v_A = 0.3 ω_A = 0.3 (2 ω) = 0.6 ω

The ratio of the linear speed between wheel A, B, and C.

vA: vB: v_C

0.6 ω : 0.6 ω : 0.3 ω

0.6 : 0.6 : 0.3

6: 6 : 3

2: 2: 1