Assume that there is no friction between m2 and the incline, that m2 = 2.4 kg, m1 = 1.2 kg, the radius of the pulley is 0.10 m, the moment of inertia of the pulley is 4.8 kg m2, and θ = 25.0°.

a) Find the combined mechanical energy of both masses the instant the masses are released.

b) What is the new gravitational potential energy of m2 once it moves 1.2 m up the incline?

c) What is the overall mechanical energy of the system once m2 moves 1.2 m up the incline?

d) What is the speed of the masses at that position?

To answer these questions, we need to understand the concepts of mechanical energy, gravitational potential energy, and conservation of energy.

a) To find the combined mechanical energy of both masses the instant they are released, we need to calculate the potential energy and kinetic energy of the system.

Potential energy (PE) is given by the formula: PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.

The initial potential energy of m2 when it is released from the incline can be calculated using: PE(m2) = m2 * g * h, where g is the acceleration due to gravity and h is the height of the incline.

Since there is no friction, all the potential energy of m2 will be converted to kinetic energy. The initial kinetic energy (KE) of m2 can be determined using the formula: KE(m2) = 0.5 * m2 * v^2, where v is the velocity of m2.

Similarly, the potential energy of m1 at that instant is m1 * g * h, and since it is on a horizontal surface, it will have no initial kinetic energy.

The combined mechanical energy (E) of both masses at that instant is the sum of the potential energy and kinetic energy of m2: E = PE(m2) + KE(m2) + PE(m1).

b) The new gravitational potential energy of m2 once it moves 1.2 m up the incline is given by PE = m2 * g * h, where h is the height. In this case, h = 1.2 m.

c) The overall mechanical energy of the system once m2 moves 1.2 m up the incline can be calculated as before, using the formula: E = PE(m2) + KE(m2) + PE(m1).

d) To find the speed of the masses at that position, we can consider the conservation of mechanical energy. The mechanical energy at the initial position (E_initial) will be equal to the mechanical energy at the new position (E_final). We can then equate these two expressions for E to solve for the final velocity (v_final) using the formula: E_initial = PE(m2_final) + KE(m2_final) + PE(m1_final).

By substituting the given values and solving the equation, we can find the final velocity of the masses.