Chapter 6: Problem 118
Think \& Calculate 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
Step by step solution
Calculate the gravitational force on the weight
Calculate the work done by the weight
Convert descent time from days to seconds
Calculate the power delivered to the clock
Discuss how to increase power
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Gravitational Force
The formula to calculate gravitational force is \( F = m \times g \), where \( m \) is the mass in kilograms and \( F \) is the force in newtons (N). For example, in our exercise, the force on a \( 4.35 \text{ kg} \) weight is calculated as \( 42.6735 \text{ N} \). This means that Earth is pulling the weight with this much force.
Understanding gravitational force is crucial for calculating how much work an object can do as it moves under this force.
Work Done
The formula for work done is \( W = F \times d \), where \( W \) is work in joules (J), \( F \) is force in newtons, and \( d \) is distance in meters. In our problem, the weight descended a distance of \( 0.760 \text{ m} \) under a force of \( 42.6735 \text{ N} \). This calculates to a total work done of \( 32.43186 \text{ J} \).
- This energy is what powers the grandfather clock, moving the components for a certain period.
Grasping this concept helps us understand how energy is used and transferred in mechanical systems, especially those driven by gravity.
Time Conversion
To convert days to seconds, remember that one day equals 86400 seconds. So, if the descent takes \( 3.25 \) days, converting this time frame involves multiplying by the number of seconds in a day:
- \( 3.25 \times 86400 = 280800 \text{ seconds} \). This long duration is then used to calculate the average power.
Properly converting time helps ensure our calculations make sense physically, as rates like power rely on this understanding.
Energy Transfer
In our grandfather clock scenario, as the weight descends, gravitational potential energy is converted into kinetic energy, and then into mechanical energy that drives the clock. This is a classic example of energy transfer.
- The weight starts with high potential energy at the top due to its position in the gravitational field. As it falls, this energy turns into motion energy, turning the gears of the clock.
Think of energy transfer as the clock 'borrowing' energy from the weight to keep the hands moving. By understanding energy transfer, we better appreciate how everyday objects function and consume energy.