Problem 65 - College Physics

Question

The two \(0.50-\mathrm{kg}\) weights of a cuckoo clock descend \(1.5 \mathrm{~m}\) in a three-day period. At what rate is their total gravitational potential energy decreased?

Step-by-Step Solution

Answer

The rate of decrease in gravitational potential energy is approximately 0.0057 J/s.

Step 1: Calculate the gravitational potential energy of the weights

The gravitational potential energy (GPE) can be calculated using the formula: \( \Delta PE = mgh \). Here, \( m \) is the mass, \( g \) is the acceleration due to gravity \( (9.8 \, \text{m/s}^2) \), and \( h \) is the change in height. For one weight: \( m = 0.50 \, \text{kg} \) and \( h = 1.5 \, \text{m} \). Thus, the GPE decrease for one weight is \( 0.50 \times 9.8 \times 1.5 \).

Step 2: Calculate the total energy decrease for both weights

Since there are two weights, the total change in gravitational potential energy is twice the change for one weight. Calculate: \( 2 \times (0.50 \times 9.8 \times 1.5) \).

Step 3: Calculate the rate of energy decrease

The change in energy occurs over three days. Convert this period into seconds by multiplying 3 days by the number of seconds in a day: \(3 \times 24 \times 60 \times 60 = 259200\) s. Divide the total energy decrease by the total seconds to find the rate of energy decrease.

Key Concepts

Physics Problem Solving

Physics problem solving involves identifying the key concepts and quantities involved in a question and applying relevant equations to solve it. Let's break down the cuckoo clock problem to understand how to determine the rate of gravitational potential energy change.
First, note the problem parameters: two weights each with a mass of 0.50 kg, and a height change of 1.5 m over 3 days.
Understanding these values is crucial, as they interact through the formula for gravitational potential energy (GPE). The formula is \( \Delta PE = mgh \). Here:

• \( m \) is the mass of the object.
• \( g \) is the acceleration due to gravity, roughly 9.8 m/s².
• \( h \) is the height the object moves through.

In our specific case, it's important to calculate the energy change for both weights and to consider the time period of 3 days, which needs conversion into seconds to find the energy change rate adequately.
This methodical approach helps solve even more complex physics problems by tackling each element stepwise.

Energy Conversion

Energy conversion is a fundamental concept in physics, where energy transitions from one form to another. In our cuckoo clock scenario, gravitational potential energy of the weights decreases as they descend, converting this energy into other forms, like kinetic energy, though it's not our focus here.
The key formula used is \( \Delta PE = mgh \) for potential energy. As the weights descend, the gravitational potential energy (GPE) decreases. With each meter they drop, a small amount of energy transforms and possibly goes on to drive the clock's mechanisms.
Understanding energy conversion allows you to see how energy flows in a system. The weights on the clock are an excellent example of mechanical energy use and conversion, moving between potential energy states. On a larger scale, this principle applies to everyday situations like how a roller coaster gains speed as it loses height, converting potential energy into kinetic energy.

Mechanical Energy

Mechanical energy in the cuckoo clock exercise involves understanding how potential and kinetic energies are aspects of mechanical energy in physical systems.
Mechanical energy is expressed as the sum of kinetic and potential energy within a system. In the case of the cuckoo clock:

• Initially, the weights have gravitational potential energy because of their raised position.
• As they fall, this potential energy changes, demonstrating the conservation and transformation of energy.

It is fundamental to note that while energy seems to disappear, it merely changes form—consistent with the principle of energy conservation. This principle states energy in a closed system remains constant, though its form can change.
This understanding highlights why calculating changes and cycles of energy within machinery, like a clock, are necessary for grasping more comprehensive concepts like mechanical systems and the interplay of forces and energy therein.