Question

The relative speed of minute-hand with respect to hour-hand is : (a) \(\left(5 \frac{1}{2}\right)^{\circ}\) per minute (b) \(\left(\frac{11}{12}\right)\) minute per minute (c) \(6^{\text {" per minute }}\) (d) both (a) and (b)

Short answer

Answer: (a) \(\left(5 \frac{1}{2}\right)^{\circ}\) per minute

Step-by-step solution

Calculate the Angular Speed of the Minute-Hand

The minute-hand completes a full circle (360 degrees) in 60 minutes. Therefore, its angular speed is: Angular speed of minute-hand = \(\frac{360}{60}\) degrees per minute = \(6^{\circ}\) per minute

Calculate the Angular Speed of the Hour-Hand

The hour-hand completes a full circle (360 degrees) in 12 hours (720 minutes). Therefore, its angular speed is: Angular speed of hour-hand = \(\frac{360}{720}\) degrees per minute = \(\frac{1}{2}^{\circ}\) per minute

Calculate the Relative Speed

The relative speed is the difference between the angular speed of the minute-hand and the hour-hand, which is: Relative speed = Angular speed of minute-hand - Angular speed of hour-hand = \(6^{\circ}\) per minute - \(\frac{1}{2}^{\circ}\) per minute = \(\left(5\frac{1}{2}\right)^{\circ}\) per minute

Comparing with the Given Options

We have found that the relative speed of the minute-hand with respect to the hour-hand is \(\left(5 \frac{1}{2}\right)^{\circ}\) per minute which matches option (a). Therefore, the answer is: (a) \(\left(5 \frac{1}{2}\right)^{\circ}\) per minute

Key concepts

Angular Speed

Understanding the concept of angular speed is crucial when solving clock problems. In simple terms, angular speed tells us how fast an object is rotating or how much angle it covers in a given amount of time.

For instance, the minute-hand of a clock completes a rotation of 360 degrees in 60 minutes. The formula for calculating angular speed is \( \text{Angular speed} = \frac{\text{Total angle of rotation}}{\text{Time taken}} \). Applying it to the minute-hand: \( \text{Angular speed of minute-hand} = \frac{360}{60} \) degrees per minute, which equates to \( 6^\circ \) per minute.

The hour-hand, on the other hand, takes much longer—720 minutes—to make a full rotation. Thus, its angular speed is only \( \text{Angular speed of hour-hand} = \frac{360}{720} \) degrees per minute, or \( \frac{1}{2}^\circ \) per minute. It's evident that the minute-hand moves much more quickly as it covers a larger angle in less time compared to the hour-hand.

Clock Arithmetic

Clock arithmetic involves understanding how the positions of the hour and minute hands relate to time and each other. It's a form of modular arithmetic where numbers wrap around after they reach a certain value—in this case, time cycles after every 12 hours.

In reference to the problem at hand, when we need to find the relative speed of one hand with respect to another, we're essentially calculating how much faster one is moving than the other. This demands subtracting the angular speed of the slower-moving hour-hand from the faster-moving minute-hand, conforming to the general arithmetic principle of difference calculation.

Using the provided calculation, we can determine that the minute-hand moves \( 6^\circ \) per minute while the hour-hand moves \( \frac{1}{2}^\circ \) per minute. Therefore, the minute-hand moves at a relative speed of \( 6^\circ - \frac{1}{2}^\circ = \left(5 \frac{1}{2}\right)^\circ \) per minute faster than the hour-hand.

Time and Distance

Although time and distance are typically used in the context of travel, within the realm of clock problems, 'distance' refers to the angle covered by the hands of the clock. The 'time' is still time as measured in seconds, minutes, or hours.

When we speak about the 'distance' a clock hand travels, we're really talking about the amount of degrees it sweeps on the clock face. So if we are to calculate 'how far' the minute hand of a clock moves in 15 minutes, we're looking for the angle it covers. That would be \( 15 \times 6^\circ = 90^\circ \) – a quarter of the clock face.

Similar principles of time and distance as known in physics apply here, but instead of meters or miles, we measure 'distance' in terms of degrees of rotation. This parallel allows us to use familiar concepts to analyze the movement of clock hands and solve related problems. In conjunction with clock arithmetic and angular speed, we can find solutions such as the relative speed between the two hands of a clock.