To find $\theta'(t)$ (the rate of change of the angle between the clock's minute and hour hands) at 2 o'clock in radians per minute, let's start by calculating the angle between the hands.

Step 1: Calculating the angle $\theta(t)$

1. Hour hand rotation:

   • The hour hand completes 1 full rotation (or $2\pi$ radians) in 12 hours.
   • Therefore, its angular speed is:
     $\omega_{\text{hour}} = \frac{2\pi}{12 \times 60} = \frac{\pi}{360} \text{ radians per minute.}$
   • At 2 o'clock, the hour hand starts at: $\theta_{\text{hour}} = \frac{2 \times 2\pi}{12} = \frac{\pi}{3} \text{ radians.}$

2. Minute hand rotation:

   • The minute hand completes 1 full rotation (or $2\pi$ radians) in 60 minutes.
   • Therefore, its angular speed is:
     $\omega_{\text{minute}} = \frac{2\pi}{60} = \frac{\pi}{30} \text{ radians per minute.}$
   • At 2 o'clock (when $t = 0$ minutes), the minute hand starts at: $\theta_{\text{minute}} = 0 \text{ radians.}$

Step 2: Finding the rate of change $\theta'(t)$

The angle between the minute and hour hands at any time $t$ (in minutes) is given by the formula: $\theta(t) = \left| \theta_{\text{minute}}(t) - \theta_{\text{hour}}(t) \right|,$ where:

• $\theta_{\text{minute}}(t) = \omega_{\text{minute}} \times t = \frac{\pi}{30} t$.
• $\theta_{\text{hour}}(t) = \frac{\pi}{3} + \omega_{\text{hour}} \times t = \frac{\pi}{3} + \frac{\pi}{360} t$.

Therefore: $\theta(t) = \left| \frac{\pi}{30} t - \left( \frac{\pi}{3} + \frac{\pi}{360} t \right) \right|$ $\theta(t) = \left| \frac{\pi}{30} t - \frac{\pi}{3} - \frac{\pi}{360} t \right|$ $\theta(t) = \left| \frac{\pi}{30} t - \frac{\pi}{360} t - \frac{\pi}{3} \right|$ $\theta(t) = \left| \frac{12\pi}{360} t - \frac{\pi}{360} t - \frac{\pi}{3} \right|$ $\theta(t) = \left| \frac{11\pi}{360} t - \frac{\pi}{3} \right|$

Step 3: Finding $\theta'(t)$

Now, take the derivative of $\theta(t)$ with respect to $t$: $\theta'(t) = \frac{d}{dt} \left( \left| \frac{11\pi}{360} t - \frac{\pi}{3} \right| \right).$

Since we want to find the derivative at 2 o'clock (which corresponds to $t = 0$), the absolute value function simplifies as: $\theta'(0) = \frac{11\pi}{360} \text{ radians per minute.}$

Final Answer

The rate of change of the angle between the clock's hands at 2 o'clock is: $\theta'(0) = \frac{11\pi}{360} \text{ radians per minute.}$

Would you like more details or have any further questions?

Here are 5 related questions for practice:

1. What is the angle between the clock's hands at 3:00?
2. How does the rate of change of the angle vary over different hours?
3. What is the rate of change of the angle at 6:00?
4. Calculate the angle between the clock's hands at 2:30.
5. How many radians does the hour hand move in 5 minutes?

Tip: Remember that the minute hand moves 12 times faster than the hour hand!