A belt connects two pulleys with radii 2 in and 4 in.. The velocity of point A on the belt is 25 ft/sec. What is the linear velocity and angular velocity for point B? What is the linear velocity and angular velocity for point C?

the 2" wheel turns twice as fast as the 4" wheel, since it has half the circumference.

To find the linear and angular velocities for point B and C on the belt, we can use the formula:

Linear velocity = Angular velocity * radius

Given the radii of the two pulleys:
Radius of pulley A = 2 inches
Radius of pulley B = 4 inches

Step 1: Calculate the angular velocity of point A.

Angular velocity of point A (ωA) = Linear velocity of point A / Radius of pulley A

Given:
Linear velocity of point A (VA) = 25 ft/sec
Radius of pulley A (rA) = 2 inches

Convert inches to feet:
rA = 2 inches * (1 ft/12 inches) = 2/12 ft = 1/6 ft

ωA = VA / rA = 25 ft/sec / (1/6 ft) = 25 * 6 ft/sec = 150 ft/sec

Therefore, the angular velocity of point A is 150 ft/sec.

Step 2: Calculate the linear and angular velocities for point B.

Since point B is on the same belt as point A, it will have the same angular velocity as point A:

Angular velocity of point B (ωB) = ωA = 150 ft/sec

To find the linear velocity of point B, we use the formula:

Linear velocity of point B (VB) = ωB * Radius of pulley B

Given:
Radius of pulley B (rB) = 4 inches

Convert inches to feet:
rB = 4 inches * (1 ft/12 inches) = 4/12 ft = 1/3 ft

VB = ωB * rB = 150 ft/sec * (1/3 ft) = 50 ft/sec

Therefore, the linear velocity of point B is 50 ft/sec and the angular velocity of point B is 150 ft/sec.

Step 3: Calculate the linear and angular velocities for point C.

Since point C is on the same belt as point B, it will have the same linear velocity as point B:

Linear velocity of point C (VC) = VB = 50 ft/sec

To find the angular velocity of point C, we use the formula:

Angular velocity of point C (ωC) = VC / Radius of pulley B

Given:
VC = 50 ft/sec
rB = 4 inches

Convert inches to feet:
rB = 4 inches * (1 ft/12 inches) = 4/12 ft = 1/3 ft

ωC = VC / rB = 50 ft/sec / (1/3 ft) = 50 * 3 ft/sec = 150 ft/sec

Therefore, the linear velocity of point C is 50 ft/sec and the angular velocity of point C is 150 ft/sec.

To find the linear and angular velocity for point B and point C, we can use the concept of angular speed.

First, let's define the terms:
- Point A is a point on the belt that has a linear velocity of 25 ft/sec.
- Point B is located on the pulley with a radius of 2 in (0.17 ft).
- Point C is located on the pulley with a radius of 4 in (0.33 ft).

To find the angular velocity for point B and point C, we can use the formula:

Angular Velocity = Linear Velocity / Radius

For point B:
The radius of the pulley at point B is 2 in (0.17 ft).
Angular velocity at point B = Linear velocity at point A / Radius at point B
Angular velocity at point B = 25 ft/sec / 0.17 ft
Angular velocity at point B ≈ 147.06 rad/sec

For point C:
The radius of the pulley at point C is 4 in (0.33 ft).
Angular velocity at point C = Linear velocity at point A / Radius at point C
Angular velocity at point C = 25 ft/sec / 0.33 ft
Angular velocity at point C ≈ 75.76 rad/sec

So, the angular velocity for point B is approximately 147.06 rad/sec, and the angular velocity for point C is approximately 75.76 rad/sec.

To find the linear velocity at point B and point C, we can use the formula:

Linear Velocity = Angular Velocity × Radius

For point B:
Linear velocity at point B = Angular velocity at point B × Radius at point B
Linear velocity at point B = 147.06 rad/sec × 0.17 ft
Linear velocity at point B ≈ 24.99 ft/sec

For point C:
Linear velocity at point C = Angular velocity at point C × Radius at point C
Linear velocity at point C = 75.76 rad/sec × 0.33 ft
Linear velocity at point C ≈ 24.99 ft/sec

Therefore, the linear velocity for both point B and point C is approximately 24.99 ft/sec, and the angular velocity for point B and point C is approximately 147.06 rad/sec and 75.76 rad/sec, respectively.