Question

A grandfather clock is powered by the descent of a \(4.35-\mathrm{kg}\) weight. (a) If the weight descends through a distance of \(0.760 \mathrm{~m}\) in \(3.25\) days, how much power does it deliver to the clock? (b) To increase the power delivered to the clock, should the time it takes for the mass to descend be increased or decreased? Explain.

Short answer

(a) The power delivered is approximately 1.16 \( \times \) 10^{-4} W. (b) Decrease descent time to increase power.

Step-by-step solution

Calculate the Work Done by the Descending Weight

The work done by the weight can be calculated using the formula for gravitational potential energy: \[ W = mgh \]where \( m \) is the mass (4.35 kg), \( g \) is the acceleration due to gravity (approximately 9.81 m/s²), and \( h \) is the height (0.760 m). Substituting the given values:\[ W = 4.35 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 0.760 \, \text{m} \approx 32.54 \, \text{J} \]

Calculate the Time in Seconds

Convert the time from days to seconds. Given that there are 3.25 days, use the conversion:\[ 1 \, \text{day} = 24 \, \text{hours} \times 60 \, \text{minutes} \times 60 \, \text{seconds} \]So, the time in seconds is:\[ t = 3.25 \, \text{days} \times 24 \, \text{h/day} \times 3600 \, \text{s/h} = 280800 \, \text{s} \]

Calculate the Power Delivered to the Clock

Power is defined as work done per unit time. Calculate the power using the formula:\[ P = \frac{W}{t} \]Substituting the known values:\[ P = \frac{32.54 \, \text{J}}{280800 \, \text{s}} \approx 1.16 \times 10^{-4} \, \text{W} \]

Determine the Effect of Changing Descent Time on Power

Power is inversely proportional to time when work is constant. Therefore, if the time it takes for the mass to descend is decreased, the power delivered to the clock will increase, and vice versa.

Key concepts

Gravitational Potential Energy

Gravitational potential energy is a kind of energy held by an object due to its position relative to Earth. Basically, when you lift something against gravity, you're storing energy in it, like winding up a toy. This energy is released when you let go, allowing the object to fall. The formula to calculate gravitational potential energy, and in extension, the work done when it moves under gravity, is:

\[ W = mgh \]
- **\( W \)** stands for work done, or the energy.- **\( m \)** is the mass of the object, measured in kilograms.- **\( g \)** is the acceleration due to gravity, with a standard value of approximately 9.81 m/s².- **\( h \)** is the height the object is moved, in meters.

For our exercise, with a 4.35 kg weight descending a height of 0.760 m, the energy transferred (or work done) was found to be 32.54 Joules.

Work Done

Work done essentially refers to how much energy is transferred when a force moves an object. It's like the effort you apply to achieve something, measured in Joules. In cases involving gravity, like lifting a weight, the work done is equivalent to the gravitational potential energy gained or lost. Here, the key points to understanding work done are:

- **Force** times **Distance**: For vertical motion under gravity, the force is \( m \cdot g \).- **Energy Transfer**: Work done is the energy transferred by the force.

When the descending weight on the clock-powered mechanism moves, gravitational potential energy turns into kinetic energy and does work on the clock mechanism. In our exercise, the work done was 32.54 Joules, achieved as the weight moves 0.760 m.

Time Conversion

Time conversion becomes very important when you have to calculate power. Here, we were given the time span in days and needed to convert it to seconds, which is a standard unit of time in physics equations. This conversion ensures that calculations like power, which needs time in seconds for accurate results, are consistent.

To get from days to seconds:

• Recognize there are 24 hours in a day.
• Know there are 60 minutes in each hour.
• And 60 seconds in each minute.

Given the time in our problem was 3.25 days, converting to seconds gives us a time value of 280,800 seconds. This conversion was crucial for subsequently calculating power.

Power Formula

Power measures how fast work is done or energy is transferred. It’s typically represented in watts (W), which corresponds to joules per second. Calculating power involves dividing the total work by the time in which this work is done. The formula used is:

\[ P = \frac{W}{t} \]
- **\( P \)**: Stands for Power.- **\( W \)**: Total work done, or energy used, in Joules.- **\( t \)**: Time in seconds over which the work is done.

In our scenario, the power delivered to the clock was roughly calculated to be approximately 1.16 × 10^{-4} watts. Interestingly, power increases when the time taken for the descent decreases, as power and time share an inverse relationship in this context.