Two masses ( 2.3 kg and 4.2 kg) connected by a massless rope hung over opposite ends of a frictionless pulley. Each mass is initially 1.50 m above the ground, and the top of the pulley is 6.5 m above the ground. What maximum height does the lighter object reach after the system is released? The answer is 3.44 m but I don't know how to get there

ryg

To find the maximum height the lighter object reaches after the system is released, we can use the principles of conservation of energy.

First, let's calculate the potential energy at the starting position. The potential energy of an object is given by the formula U = mgh, where U is the potential energy, m is the mass of the object, g is the acceleration due to gravity, and h is the height above a reference point.

For the 2.3 kg mass:
U1 = 2.3 kg × 9.8 m/s² × 1.5 m = 33.99 J

For the 4.2 kg mass:
U2 = 4.2 kg × 9.8 m/s² × 1.5 m = 73.47 J

Next, let's consider the change in potential energy as the masses move. As the masses descend, their potential energy is converted into kinetic energy. At the maximum height, the kinetic energy is zero, and all the initial potential energy is transformed into potential energy at that height.

The potential energy at the maximum height for the lighter object can be determined by subtracting the potential energy it had at the start from the total potential energy:

U_max = U2 - U1 = 73.47 J - 33.99 J = 39.48 J

Finally, we can find the height corresponding to this potential energy by rearranging the potential energy formula:

U_max = mgh_max

Since the mass cancels out, we can calculate the height (h_max) using:

h_max = U_max / (mg)

For the lighter object, assuming a mass of 2.3 kg:

h_max = 39.48 J / (2.3 kg × 9.8 m/s²) = 1.74 m

However, this value is measured from the reference point (the ground). To find the maximum height from the starting position (1.5 m above the ground), we need to add the initial height:

Maximum height from the starting position = h_max + 1.5 m = 1.74 m + 1.5 m = 3.24 m

Therefore, the maximum height the lighter object reaches after the system is released is approximately 3.24 m.