A hockey puck of mass m = 120 g is attached to a string that passes through a hole in the center of a table, as shown in the figure below. The hockey puck moves in a circle of radius r = 0.50 m. Tied to the other end of the string, and hanging vertically beneath the table, is a mass M = 0.50 kg. Assuming the tabletop is perfectly smooth, what speed must the hockey puck have if the mass M is to remain at rest?
To find the speed required for the hockey puck in order for the mass M to remain at rest, we need to consider the forces acting on both objects.
First, let's analyze the forces acting on the mass M. Since it is hanging vertically and at rest, the only force acting on it is its weight (mg), pointing downwards. This force is balanced by the tension in the string.
Now, let's examine the forces acting on the hockey puck. There are two forces: the tension in the string pulling towards the center of the circular path, and the gravitational force pulling downwards.
To keep the mass M at rest, the tension in the string must exactly balance its weight (mg). Therefore, we can set up the following equation:
Tension in string = mg
Now, let's consider the forces acting on the hockey puck. The tension in the string provides the necessary centripetal force to keep the puck moving in a circle.
The centripetal force is given by the equation:
Centripetal force = (mass of the puck) x (velocity of the puck)^2 / (radius of the circle)
In this case, the centripetal force is equal to the tension in the string. So we can set up the following equation:
Tension in string = (mass of the puck) x (velocity of the puck)^2 / (radius of the circle)
Since we want to find the speed of the puck, we rearrange and solve for velocity:
(velocity of the puck)^2 = (tension in the string) x (radius of the circle) / (mass of the puck)
Substituting the tension in the string with the weight of mass M (mg), we get:
(velocity of the puck)^2 = (mg) x (radius of the circle) / (mass of the puck)
Now we can plug in the given values:
mass of the puck (m) = 120 g = 0.12 kg
radius of the circle (r) = 0.50 m
mass of the hanging mass (M) = 0.50 kg
acceleration due to gravity (g) = 9.8 m/s^2
(velocity of the puck)^2 = (0.50 kg) x (9.8 m/s^2) x (0.50 m) / (0.12 kg)
Simplifying the equation:
(velocity of the puck)^2 = 20.41 m^2/s^2
Taking the square root of both sides:
velocity of the puck = 4.52 m/s
Therefore, the hockey puck must have a speed of 4.52 m/s for the mass M to remain at rest.