Question

A particle moves without friction. At point A the particle has a kinetic energy of \(12 \mathrm{~J}\); at point B the particle is momentarily at rest, and the potential energy of the system is \(25 \mathrm{~J}\); at point \(\mathrm{C}\) the potential energy of the system is \(5 \mathrm{~J}\). (a) What is the potential energy of the system when the particle is at point \(A\) ? (b) What is the kinetic energy of the particle at point \(C\) ?

Short answer

(a) 13 J; (b) 20 J

Step-by-step solution

Understanding Energy Conservation

The energy conservation principle states that total mechanical energy (sum of kinetic and potential energy) in a system remains constant if only conservative forces act on it. For this problem, assume no energy is lost due to friction.

Calculate Total Mechanical Energy at A

At point A, the kinetic energy is given as 12 J. Let the potential energy at A be \(U_A\), thus the total mechanical energy at A is \(E_A = K_A + U_A = 12 \text{ J} + U_A\).

Calculate Total Mechanical Energy at B

At point B, the particle comes to rest, so its kinetic energy is 0 J. The potential energy is given as 25 J. Hence, total mechanical energy at B is \(E_B = 0 \text{ J} + 25 \text{ J} = 25 \text{ J}\).

Equate Total Mechanical Energy at A and B

Since there is no energy loss, equate the total mechanical energy at points A and B: \[E_A = E_B\] \[12 \text{ J} + U_A = 25 \text{ J}\]. Solve for \(U_A\): \(U_A = 25 \text{ J} - 12 \text{ J} = 13 \text{ J}\).

Total Mechanical Energy at C

Using conservation of energy, the total mechanical energy at C must also be \(E_C = E_A = 25 \text{ J}\). Given the potential energy at C is 5 J, the equation is \(E_C = K_C + 5 \text{ J}\).

Solve for Kinetic Energy at C

Substitute the value of total mechanical energy at C (25 J) to find the kinetic energy: \[K_C + 5 \text{ J} = 25 \text{ J}\] Solve for \(K_C\): \(K_C = 25 \text{ J} - 5 \text{ J} = 20 \text{ J}\).

Key concepts

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. It is calculated using the formula:\[ K = \frac{1}{2} m v^2 \]where \( m \) is the mass and \( v \) is the velocity of the object. In simpler terms, the faster something is moving, the more kinetic energy it has. In our exercise, when the particle is at point A, it has a kinetic energy of 12 J. This means that at point A, the object is in motion. The energy it has at this moment is only due to its speed, as kinetic energy depends only on the velocity of the object. Since no external forces like friction are acting here, this energy remains part of the system's total mechanical energy as it moves along. Understanding kinetic energy helps in analyzing how the energy of a moving object changes as it encounters different potential energies during its path.

Potential Energy

Potential energy is the energy stored in an object due to its position or arrangement. It is often associated with the object's height above the ground or the configuration of a system like a spring. The formula for gravitational potential energy is:\[ U = mgh \]where \( m \) is mass, \( g \) is the gravitational acceleration, and \( h \) is the height. This formula helps in knowing how much energy an object has stored because of its position. In our particular scenario, the particle at point B has a potential energy of 25 J, indicating it is at a specific height connected to this energy value. Knowing the potential energy helps us calculate energies at different points without losing track of the total mechanical energy. For instance, the energy remains 25 J at point B even though kinetic energy is zero.

Mechanical Energy

Mechanical energy is the sum of kinetic and potential energy in a system. It is a crucial concept in understanding energy conservation, as it remains constant in conservative systems (systems without non-conservative forces like friction). The formula for mechanical energy is:\[ E = K + U \]where \( E \) is the total mechanical energy, \( K \) is the kinetic energy, and \( U \) is the potential energy. Our exercise illustrates how the mechanical energy at any point in the system's path should equal the total energy elsewhere, assuming no energy losses. For instance, at point A, the sum of kinetic and potential energy was computed to be 25 J, similarly at point B, despite the different energy forms. Understanding mechanical energy helps assess how energy transitions between kinetic and potential forms while affirming that total energy in the system stays constant throughout. This principle underlies numerous physical processes and technology applications.