A 2 kg mass on a horizontal friction-free air track is accelerated by a string attached to a 115 kg mass hanging vertically from a pulley as shown. Compare the accelerations when the masses are interchanged, that is, for the case where the 115 kg mass dangles over the pulley, and then for the case where the 2 kg mass dangles over the pulley.

Question

A 2 kg mass on a horizontal friction-free air track is accelerated by a string attached to a 115 kg mass hanging vertically from a pulley as shown. Compare the accelerations when the masses are interchanged, that is, for the case where the 115 kg mass dangles over the pulley, and then for the case where the 2 kg mass dangles over the pulley.

Answer 1

(a)

total mass = (M+m)
Total force = Mg
Acceleration = Mg/(M+m)
(b)
Acceleration = mg/(M+m)

Answer 2

To compare the accelerations in the two cases, we need to apply Newton's second law of motion. This law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

Let's consider the case where the 115 kg mass hangs vertically from the pulley. In this case, the tension in the string will be equal to the weight of the hanging mass. The weight of an object is given by the formula W = m * g, where m is the mass and g is the acceleration due to gravity (approximately 9.8 m/s^2).

The weight of the 115 kg mass is W = 115 kg * 9.8 m/s^2 = 1127 N.

Since the system is friction-free, the tension in the string will be the only force acting on the 2 kg mass. Therefore, we can write:

Tension = mass * acceleration

1127 N = 2 kg * acceleration

Simplifying the equation gives us the acceleration when the 115 kg mass dangles over the pulley:

acceleration = 1127 N / 2 kg = 563.5 m/s^2

Now let's consider the case where the 2 kg mass dangles over the pulley. In this case, the tension in the string will be equal to the weight of the 2 kg mass, which is:

Weight = mass * g = 2 kg * 9.8 m/s^2 = 19.6 N

Again, the tension in the string will be the only force acting on the 115 kg mass. So we can write:

Tension = mass * acceleration

19.6 N = 115 kg * acceleration

Simplifying the equation gives us the acceleration when the 2 kg mass dangles over the pulley:

acceleration = 19.6 N / 115 kg = 0.17 m/s^2

Therefore, when the masses are interchanged, the acceleration of the system with the 115 kg mass over the pulley is 563.5 m/s^2, while the acceleration with the 2 kg mass over the pulley is 0.17 m/s^2.