Question

A ball of mass \(12.2 \mathrm{~g}\) is dropped from rest at a height of \(76 \mathrm{~cm}\) above the surface of oil that fills a barrel to a depth of \(55 \mathrm{~cm}\). The ball reaches the bottom of the barrel with a speed of \(1.48 \mathrm{~m} / \mathrm{s}\). (a) Neglecting air resistance, find the speed of the ball when it enters the oil. \((b)\) What is the change in the internal energy of the system of ball + oil?

Short answer

The speed of the ball when it enters the oil is determined in Step 1. The change in the internal energy of the system of ball+oil is then calculated in Step 3 using the kinetic energies at the starting and end points.

Step-by-step solution

Calculate the initial speed

Use the equation of motion \(v^2 = u^2 + 2gs\) to find the speed 'v' of the ball when it enters the oil. Here, 'u' is the initial speed which is 0 (since the ball is dropped), 'g' is the acceleration due to gravity (9.81 m/s²), and 's' is the distance the ball travels before hitting the oil (in meters). It's important to change the distance from cm to m by dividing by 100, so \(s = 76/100 = 0.76m\). Now, let's substitute these values into the equation to find 'v'.

Calculate the change in internal energy

To find the change in internal energy of the system of ball + oil, first we will need to calculate the initial and final kinetic energy. The initial kinetic energy of the system, considering the ball about to enter the oil, is given by \(KE_i = 0.5 * m * v^2\), where m is the mass of the ball and v is the speed calculated in the previous step. The final kinetic energy of the system when the ball hits the bottom of the barrel with a speed of 1.48 m/s is given by \(KE_f = 0.5 * m * (1.48)^2\). The change in internal energy is then given by \(\Delta E = KE_f - KE_i\).

Calculate the final values

Use the values of mass \(m = 12.2 / 1000 = 0.0122 kg\) (converted from g to kg), and the speeds we calculated for 'v' in step one and from the problem statement to find the values of KE_i and KE_f. Subtract KE_i from KE_f to get the change in internal energy.

Key concepts

Kinematics

Kinematics is the branch of physics that deals with understanding the motion of objects without considering the forces causing them. In this particular problem, a ball is dropped from a certain height, and we need to understand the kinematic equations to find its speed at different points during its fall. One helpful equation is the motion equation \( v^2 = u^2 + 2gs \), where:

• \( v \) is the final velocity.
• \( u \) is the initial velocity (\( 0 \) for the dropped ball).
• \( g \) is the acceleration due to gravity, typically \( 9.81 \text{ m/s}^2 \).
• \( s \) is the displacement or change in position.

By substituting these values and observing the motion of the ball from the drop to entering the oil, we can effectively determine the unknown speed just as the ball hits the oil.

Energy Conservation

One of the core principles in physics is the conservation of energy. This principle states that energy cannot be created or destroyed; it only transforms from one form to another. When an object falls, its potential energy (determined by its height) converts into kinetic energy (movement energy).
In this exercise, when the ball falls from its initial height, it starts with potential energy and changes into kinetic energy as it drops. When the ball first touches the oil, most of that potential energy has now become kinetic energy. By calculating kinetic energy at these points, we can observe this energy transformation.

Internal Energy

Internal energy refers to the energy stored within a system. In this problem, the ball and oil system's internal energy changes because of energy transformations and transfers, especially during interactions, such as when the ball enters the oil.
To calculate this, we need to find the change in kinetic energy of the ball before and after entering the oil. The difference in kinetic energies represents the amount of energy transferred to the internal energy of the system, often appearing as thermal energy or other forms. This change in internal energy tells us how much energy was added or lost from the system when the ball interacts with the oil.

Motion Equations

Motion equations help us describe an object's movement through space over time. They are crucial for solving problems involving initial and final speeds, displacement, and acceleration.
For instance, in this problem, the equation \( v^2 = u^2 + 2gs \) is a motion equation used to find the speed when an object falls from a height.

• It assumes constant acceleration, which is typical for gravity-driven scenarios.
• This equation is highly beneficial for determining kinetic factors like velocity and displacement.

Understanding how to apply these equations allows us to deal with problems involving different stages of motion, like determining the speed of the ball as it moves through various mediums in this scenario.