# Wheels connected by belts – problems and solutions

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

If RA = 10 cm, RB = 4 cm, and RC = 40 cm, then the ratio of the angular velocity of wheel A and wheel C is …

Known : Radius of wheel A (rA) = 10 cm

Radius of wheel B (rB) = 4 cm

Radius of wheel C (rC) = 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 (360o), during the same time interval the wheel A not yet rotates one circle (360o). 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 (vC) equal to the linear speed of the edge of the wheel A (vA).

vA = vC

rA ωA = rC ωC

10 ωA = 40 ωC

ωA / ωC = 40 / 10

ωA / ωC = 4 / 1

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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 (rA) = 30 cm = 0.3 meters Radius of wheel B (rB) = 60 cm = 0.6 meters

Radius of wheel C (rC) = 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 (360o), wheel B not yet around one circle (360o). 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 (vA) is equal to the linear velocity of the edge of the wheel B (vB).

The linear speed of the edge of wheel A :

vA = rA ωA = 0.3 ωA

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 (360o) than during the same time interval, wheel C also around one circle (360o). Since it rotates together, then the angular speed of wheel B (ωB) is equal to the angular speed of wheel C (ωC) = ω. But the linear speed of wheel B (vB) is not equal to the linear speed of wheel C (vC)

The linear speed of the edge of wheel B :

vB = rB ωB = 0.6 ωB = 0.6 ω

The linear speed of the edge of wheel C :

vC = rC ωC = 0.3 ωC = 0.3 ω

The linear speed of the edge of wheel A (vA) same as the linear speed of the edge of wheel B (vB)

vA = vB

0.3 ωA = 0.6 ω

ωA = 0.6 ω / 0.3

ωA = 2 ω

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

vA = 0.3 ωA = 0.3 (2 ω) = 0.6 ω

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

vA: vB: vC

0.6 ω : 0.6 ω : 0.3 ω

0.6 : 0.6 : 0.3

6: 6 : 3

2: 2: 1

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