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 whee****l 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}

**T****he linear speed of the edge of the whee****l 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 wh****e****el 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