A 2 kg mass on a horizontal friction-free air track is accelerated by a string attached to a 100 kg mass hanging vertically from a pulley as shown. Compare the accelerations when the masses are interchanged, that is, for the case where the 100 kg mass dangles over the pulley, and then for the case where the 2 kg mass dangles over the pulley.

acceleration with 100 kg mass dangling g
acceleration with 2 kg mass dangling g

What does this indicate about the maximum acceleration of such a system of masses?

To compare the accelerations when the masses are interchanged, we can use Newton's second law of motion, which states that the force acting on an object is equal to its mass multiplied by its acceleration (F = ma). In this case, the force acting on both masses is the tension in the string.

When the 100 kg mass is dangling over the pulley, the tension in the string is equal to the weight of the 100 kg mass, which is given by the formula F = mg, where m is the mass and g is the acceleration due to gravity. Therefore, the tension is T = (100 kg)(9.8 m/s^2) = 980 N.

Using Newton's second law, we can set up an equation for the acceleration of the 2 kg mass when the 100 kg mass is dangling:

T - (2 kg)(a) = (2 kg)(a)

Simplifying the equation, we find:

980 N - 2 kg(a) = 2 kg(a)
980 N = 4 kg(a)
a = 245 m/s^2

So, when the 100 kg mass is dangling, the 2 kg mass undergoes an acceleration of 245 m/s^2.

Now, let's consider the case where the 2 kg mass is dangling over the pulley. Since the same string connects the masses, the tension in the string will be the same as before, which is 980 N.

Using Newton's second law again, we can set up an equation for the acceleration of the 100 kg mass when the 2 kg mass is dangling:

T - (100 kg)(g) = (100 kg)(g)

Simplifying the equation, we find:

980 N - 100 kg(g) = 100 kg(g)
980 N = 200 kg(g)
g = 4.9 m/s^2

So, when the 2 kg mass is dangling, the 100 kg mass undergoes an acceleration of 4.9 m/s^2.

Comparing the two accelerations, we see that when the 2 kg mass is dangling, the system has a lower acceleration compared to when the 100 kg mass is dangling. Therefore, this indicates that the maximum acceleration of such a system of masses is greater when the larger mass is hanging vertically from the pulley.