Question

Suppose we have two masses M attached to a string looped over a pulley. The mass on the left is lying on the floor, while the right one is suspended at y = h1. Suppose that at y = d + h1 we have an extra mass m attached to the string (above M). The string is assumed to be stretched vertically and we eventually release the pulley.

(a) Obtain the acceleration for all the masses
(b) Using this acceleration, what is the speed at which the lower mass at the right hits the floor?
(c) If instead of having the right masses m and M separated by a string, we have them glued to each other, show whether the calculation/results would be changed?

Answer 1

To calculate the acceleration for all the masses in this system, we can use Newton's second law of motion. Once we have the acceleration, we can use it to calculate the speed at which the lower mass on the right hits the floor. If the masses m and M are glued together instead of being separated by a string, the calculation and results would be different.

Step-by-step explanation:

To calculate the acceleration for all the masses in this system, we can use Newton's second law of motion, which states that the net force on an object is equal to the mass of the object multiplied by its acceleration. In this case, we have two masses connected by a string. Let's assume that M is the mass on the left and m is the hanging mass on the right.

The net force on M is equal to Tension - Weight of M, where Tension is the tension in the string and Weight of M is equal to M multiplied by the acceleration due to gravity, g. On the other hand, the net force on m is equal to Tension - Weight of m. Since the two masses are connected by the same string, the tension is the same for both masses.

Using the formulas mentioned above, we can write the following equations:
Net force on M = Tension - Weight of M = M * a
Net force on m = Tension - Weight of m = m * a

where a is the acceleration of the masses. We can solve these equations to find the acceleration:

a = (Tension - M * g) / (M + m)

Once we have the acceleration, we can use it to calculate the speed at which the lower mass on the right hits the floor. We can use the kinematic equation:
v^2 = u^2 + 2 * a * s

where v is the final velocity, u is the initial velocity (which is zero since the mass starts from rest), a is the acceleration, and s is the distance the mass travels (which is d + h1). We can rearrange the equation to solve for v:

v = sqrt(2 * a * (d + h1))

If the masses m and M are glued together instead of being separated by a string, the calculation and results would be different. In this case, the two masses would have the same acceleration, since they are now considered as one single object. The acceleration can be calculated using the same formula as before, but with M + m as the total mass.

Now, let's calculate the acceleration for all the masses and the speed at which the lower mass on the right hits the floor.