Problem 55 - University Physics with Modern Physics

Question

A system of two paint buckets connected by a lightweight rope is released from rest with the \(12.0-\mathrm{kg}\) bucket 2.00 \(\mathrm{m}\) above the floor (Fig. 7.36\() .\) Use the principle of conservation of energy to find the speed with which this bucket strikes the floor. You can ignore friction and the mass of the pulley.

Step-by-Step Solution

Answer

The final speed of the bucket is approximately 6.26 m/s.

Step 1: Understand the Problem

We have a system consisting of two buckets connected by a rope, with one of the buckets weighing 12.0 kg and initially positioned 2.00 meters above the ground. The goal is to find the speed of this bucket when it hits the ground, assuming no energy losses due to friction or the pulley’s mass.

Step 2: Outline the Conservation of Energy Principle

The principle of conservation of energy states that the total energy in an isolated system remains constant. Here, the potential energy of the elevated bucket will be completely converted into the kinetic energy of the buckets when the 12.0 kg bucket reaches the floor.

Step 3: Write the Energy Conservation Equation

Initially, the system has gravitational potential energy, given by \( U_i = mgh \), where \( m = 12.0 \text{ kg} \) and \( h = 2.00 \text{ m} \). The bucket's kinetic energy when it hits the floor is given by \( K_f = \frac{1}{2}mv_f^2 \). Thus, the energy conservation equation is: \[ mgh = \frac{1}{2}mv_f^2 \]

Step 4: Solve for Final Speed

Solve the conservation equation for the final speed \( v_f \). Since the mass \( m \) appears on both sides of the equation, it cancels out. This gives us:\[ gh = \frac{1}{2}v_f^2 \]Rearranging and solving for \( v_f \), we get:\[ v_f = \sqrt{2gh} = \sqrt{2 \times 9.81 \text{ m/s}^2 \times 2.00 \text{ m}} \]

Step 5: Calculate the Final Speed

Calculate the final speed using the values for \( g \) and \( h \):\[ v_f = \sqrt{2 \times 9.81 \times 2.00} = \sqrt{39.24} \approx 6.26 \text{ m/s} \]

Key Concepts

Gravitational Potential Energy

Gravitational potential energy is an important concept in physics that refers to the energy stored in an object due to its height above a reference point. In our context, it is tied to the 12 kg paint bucket suspended 2 meters above the floor. This type of energy is given by the formula: \( U = mgh \), where \( m \) is the mass of the object, \( g \) is the acceleration due to gravity (approximated to be 9.81 m/s² on Earth), and \( h \) is the height above the reference point.

In the given exercise: - The mass \( m \) is 12.0 kg. - The height \( h \) is 2.00 m. - Thus, the gravitational potential energy is calculated as \( U = 12.0 \times 9.81 \times 2.00 \).

This potential energy represents how much work the gravitational force can do on the bucket as it falls, transforming that energy into other forms, such as kinetic energy. Understanding gravitational potential energy helps in solving many real-world physics problems involving heights and brought into play whenever objects are lifted to a certain height.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. When the paint bucket, initially at rest, begins to fall, its potential energy is converted into kinetic energy.

Kinetic energy is expressed by the formula: \( K = \frac{1}{2}mv^2 \), where \( m \) is the mass of the object and \( v \) is its velocity.

For the falling bucket: - The mass \( m \) remains 12.0 kg. - As it falls, it gains speed, and the kinetic energy increases until it reaches the floor. - The formula implies that even a small increase in velocity results in a significant increase in kinetic energy, due to the squaring of \( v \).

In the exercise, conservation of energy principle allows us to equate the initial potential energy to the final kinetic energy, leading to calculating the speed of the bucket when it strikes the floor. This speed is crucial for determining the kinetic energy at that moment.

Energy Conversion

Energy conversion is a process of changing one form of energy into another, and it plays a vital role in understanding how systems behave under the principle of conservation of energy.

In the case of the paint buckets, initially, the system possesses gravitational potential energy due to the elevated position of one bucket. As the bucket begins to fall, this stored potential energy is converted into kinetic energy.

Here is how the conversion process unfolds: - At the start, with the bucket elevated, the system's energy is entirely potential. - As the bucket falls, the potential energy decreases while the kinetic energy increases. - If there were no other forces (like friction) acting on the system, the potential energy would be completely converted into kinetic energy by the time the bucket hits the ground.

The principle of energy conversion exemplifies how energy remains within a system but changes its form to accomplish motion or other effects. Understanding this conversion aids in predicting the behavior of dynamic systems and is foundational in physics education and practical applications.