Question

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

(a) 0.0001155 Watts. (b) Decrease the time for descent.

Step-by-step solution

Calculate the gravitational force on the weight

To begin, we need to find the gravitational force exerted on the weight. This is given by the formula: \[ F = m imes g \] where \( m = 4.35 \) kg is the mass of the weight and \( g = 9.81 \) m/s² is the acceleration due to gravity. Therefore, \[ F = 4.35 \times 9.81 = 42.6735 \text{ N} \].

Calculate the work done by the weight

The work done by the weight as it descends is given by the equation: \[ W = F \times d \] where \( d = 0.760 \) m is the distance. Substituting the values, we have: \[ W = 42.6735 \times 0.760 = 32.43186 \text{ J} \].

Convert descent time from days to seconds

The descent time is given in days, so we need to convert it into seconds. One day has 86400 seconds, so:\[ 3.25 \text{ days} = 3.25 \times 86400 = 280800 \text{ seconds} \].

Calculate the power delivered to the clock

Power is defined as the work done per unit time. Using the formula: \[ P = \frac{W}{t} \] where \( W = 32.43186 \text{ J} \) is the work done and \( t = 280800 \text{ seconds} \) is the time. So, the power is: \[ P = \frac{32.43186}{280800} \approx 0.0001155 \text{ Watts} \].

Discuss how to increase power

Power can be increased by either increasing the work done or decreasing the time taken. Since the work done is determined by the weight and distance, we should focus on time. To increase power, the time for descent should be decreased so that the same amount of work is done in a shorter period.

Key concepts

Gravitational Force

Gravitational force is the attractive force that pulls two masses towards each other. On Earth, this force is the pull that the planet exerts on objects at its surface. The strength of this force is determined by two factors: the mass of the object and the acceleration due to gravity (denoted as \( g \)). In most physics problems near Earth's surface, \( g \) is a constant approximately equal to \( 9.81 \text{ m/s}^2 \). This value shows how fast an object will speed up when falling freely under Earth's gravity.

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

Work is the measure of energy transfer when a force moves an object over a distance. It's a way of quantifying how much energy is being used or transferred during this motion. When it comes to gravitational forces, the work done is determined by how far an object moves in the direction of the force.

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

Time conversion is an important step in many physics problems, especially if units differ from the standard scientific units, which are seconds (s). In our case, the problem initially gives time in days, but for accurate power calculations, we need it in seconds.

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

Energy transfer is a key concept in physics, describing how energy moves from one part of a system to another or transforms into different forms. In mechanical systems, energy often transforms between potential, kinetic, and other energetic forms.

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.