An air puck of mass m1 = 0.24 kg is tied to a string and allowed to revolve in a circle of radius R = 1.4 m on a frictionless horizontal table. The other end of the string passes through a hole in the center of the table, and a mass of m2 = 1.3 kg is tied to it (see the figure below). The suspended mass remains in equilibrium while the puck on the tabletop revolves.
(a) What is the tension in the string?
N
(b) What is the horizontal force acting on the puck?
N
(c) What is the speed of the puck?
m
To find the answers to these questions, we can use principles of rotational motion and forces.
(a) To find the tension in the string, we need to consider the forces acting on the system. The only vertical force is the weight of the hanging mass m2, which is equal to m2 * g, where g is the acceleration due to gravity (approximately 9.8 m/s^2). In equilibrium, the tension in the string must balance this weight, so the tension in the string is equal to m2 * g.
(b) To find the horizontal force acting on the puck, we need to consider the centripetal force required to keep it moving in a circle. The centripetal force is given by the equation Fc = (m1 * v^2) / R, where m1 is the mass of the puck, v is its velocity, and R is the radius of the circular path. Since there is no friction, the centripetal force is provided solely by the tension in the string. So, the horizontal force acting on the puck is equal to the tension in the string.
(c) To find the speed of the puck, we can use the centripetal force equation again and equate it to the tension in the string. Fc = (m1 * v^2) / R. Substituting the tension in the string as the centripetal force, we have T = (m1 * v^2) / R. Rearranging the equation, we can solve for v: v = sqrt((T * R) / m1).
Given the values of m1 = 0.24 kg, R = 1.4 m, and m2 = 1.3 kg, we can now calculate the answers.
(a) Tension in the string = m2 * g = 1.3 kg * 9.8 m/s^2 = 12.74 N
(b) Horizontal force acting on the puck = Tension in the string = 12.74 N
(c) Speed of the puck = sqrt((T * R) / m1) = sqrt((12.74 N * 1.4 m) / 0.24 kg) ≈ 6.08 m/s