Let's analyze the problem step by step.

Given:

• The radius of the wheel is $r = 14$ cm.
• The wheel completes one revolution every 7 seconds.

Part (a) Angular Velocity:

Angular velocity ($\omega$) is defined as the angle (in radians) per unit time. Since one complete revolution corresponds to an angle of $2\pi$ radians, we can calculate the angular velocity.

$\omega = \frac{\theta}{t}$

Where:

• $\theta$ is the angle of rotation in radians (for one revolution, $\theta = 2\pi$ radians),
• $t$ is the time taken for one revolution (given as 7 seconds).

Thus, the angular velocity is:

$\omega = \frac{2\pi \, \text{radians}}{7 \, \text{seconds}} \approx 0.8976 \, \text{radians/second}$

Part (b) Linear Velocity:

The linear velocity ($v$) is related to the angular velocity by the equation:

$v = r \times \omega$

Where:

• $r$ is the radius of the wheel (14 cm),
• $\omega$ is the angular velocity (calculated above as $\omega = 0.8976 \, \text{radians/sec}$).

Substituting the values:

$v = 14 \, \text{cm} \times 0.8976 \, \text{radians/sec} \approx 12.57 \, \text{cm/sec}$

Final Answers:

• (a) The angular velocity of the wheel is approximately $0.8976 \, \text{radians/sec}$.
• (b) The linear velocity of the hamster running is approximately $12.57 \, \text{cm/sec}$.

Would you like a more detailed breakdown or further assistance?

Related Questions:

1. How would the angular velocity change if the wheel took 5 seconds to complete one revolution?
2. What would happen to the linear velocity if the radius of the wheel were doubled?
3. How do we calculate the distance traveled by the hamster in one minute?
4. If the hamster ran faster and completed a revolution in 5 seconds, how would the angular velocity change?
5. How do we convert angular velocity from radians per second to degrees per second?

Tip:

To visualize angular and linear velocities, remember that angular velocity describes how fast something rotates, while linear velocity tells you how fast a point on the edge of that object moves.