A belt connects two pulleys with radii 3 in. and 5 in. as shown in the accompanying diagram. The velocity of point A on the belt is 25 ​ft/sec. What is the linear velocity and the angular velocity for point​ B? What is the linear velocity and the angular velocity for point​ C? 
​Hint: Every point on the belt is moving at the same speed.

We are given a system with two pulleys of radii 3 inches and 5 inches, connected by a belt. The velocity of point A on the belt is 25 ft/sec, and we need to find the **linear and angular velocities** for points B and C on the pulleys.

### Key Insights:
- **Linear velocity (v)**: The speed at which a point on the rim of the pulley moves. For both pulleys, every point on the belt moves at the same **linear velocity** (given as 25 ft/sec).
- **Angular velocity (ω)**: The rate at which the pulley rotates. It is related to the linear velocity by the formula:

\[
v = r \cdot \omega
\]
where \( r \) is the radius and \( \omega \) is the angular velocity in radians per second.

### Step-by-Step Solution:

#### 1. Find the linear velocity for point B:
Since the velocity of point A on the belt is given as 25 ft/sec, and point B is also on the belt, the **linear velocity** at point B is the same as at point A:
\[
v_B = 25 \text{ ft/sec}
\]

#### 2. Find the angular velocity for point B:
Using the formula \( v = r \cdot \omega \), where \( r_B = 5 \text{ in} = \frac{5}{12} \text{ ft} \) (convert inches to feet):
\[
25 = \frac{5}{12} \cdot \omega_B
\]
Solve for \( \omega_B \):
\[
\omega_B = \frac{25 \times 12}{5} = 60 \text{ radians per second}
\]

#### 3. Find the linear velocity for point C:
As mentioned earlier, the linear velocity is the same across the entire belt. Hence, the **linear velocity** at point C is:
\[
v_C = 25 \text{ ft/sec}
\]

#### 4. Find the angular velocity for point C:
Using the same formula \( v = r \cdot \omega \), with \( r_C = 3 \text{ in} = \frac{3}{12} \text{ ft} \):
\[
25 = \frac{3}{12} \cdot \omega_C
\]
Solve for \( \omega_C \):
\[
\omega_C = \frac{25 \times 12}{3} = 100 \text{ radians per second}
\]

### Summary of Results:
- **Point B** (on the 5-inch pulley):
  - Linear velocity: \( 25 \text{ ft/sec} \)
  - Angular velocity: \( 60 \text{ rad/sec} \)

- **Point C** (on the 3-inch pulley):
  - Linear velocity: \( 25 \text{ ft/sec} \)
  - Angular velocity: \( 100 \text{ rad/sec} \)

Do you want more details on how the equations were derived, or any clarifications? Here are some questions you might explore next:

1. How does the radius of a pulley affect its angular velocity?
2. What would happen to the angular velocity if the belt velocity increased?
3. How can we calculate the number of revolutions per second for each pulley?
4. What is the relationship between linear and angular velocities for objects in circular motion?
5. How does the belt's tension affect the pulley's motion?

**Tip**: Always convert all units to the same system (like feet or inches) before solving equations.

A belt connects two pulleys with radii 3 in. and 5 in. as shown in the accompanying diagram. The velocity of point A on the belt is 25 ft/sec. What is the linear velocity and the angular velocity for point B? What is the linear velocity and the angular velocity for point C?

Learn how to calculate the linear and angular velocities of two points on pulleys connected by a belt. The problem involves pulleys with radii of 3 inches and 5 inches, and a belt moving at 25 ft/sec. This step-by-step solution covers circular motion concepts and angular velocity calculations, suitable for Grades 10-12.

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