A 1.42 kg ball is connected by means of two massless strings, each of length L = 1.29 m, to a vertical, rotating rod. The strings are tied to the rod with separation d = 1.29 m and are taut. The tension in the upper string is 48.0 N. What are (a) the tension in the lower string, (b) the magnitude of the net force Upper F Overscript right-arrow EndScripts Subscript net on the ball, and (c) the speed of the ball?
a.) 32.8N
b.) 1.7m
c.) 12m/s
To find the answers to these questions, we can use Newton's laws of motion and the principles of rotational dynamics.
(a) The tension in the lower string:
Since the ball is in equilibrium, the net force acting on it must be zero. The tension in the upper string is given as 48.0 N, and since the strings are taut and connected to the same rotating rod, the tension in the lower string should also be the same, 48.0 N.
(b) The magnitude of the net force on the ball:
The net force on the ball can be calculated by considering the tension in the strings and the gravitational force acting on the ball. The gravitational force is given by the equation Fg = mg, where m is the mass of the ball and g is the acceleration due to gravity (9.8 m/s^2).
Fg = (1.42 kg) * (9.8 m/s^2) = 13.916 N
Since the ball is in equilibrium, the net force acting on it is zero. This means that the tension in the upper string and the gravitational force must cancel each other out, resulting in:
Net force on the ball = Tension in the upper string - Gravitational force
Net force on the ball = 48.0 N - 13.916 N = 34.084 N
Therefore, the magnitude of the net force on the ball is 34.084 N.
(c) The speed of the ball:
To find the speed of the ball, we can use the concept of centripetal force. The ball is connected to a rotating rod, which means it is moving in a circular path. The tension in the upper string provides the centripetal force that keeps the ball in circular motion.
The centripetal force can be calculated using the formula Fc = (mv^2) / r, where m is the mass of the ball, v is its speed, and r is the radius of the circular path.
In this case, the radius can be calculated as the sum of the lengths of the two strings:
Radius (r) = 2L + d
r = 2(1.29 m) + 1.29 m = 3.87 m
Since the tension in the upper string is providing the centripetal force, we can equate it to the centripetal force:
Tension in the upper string = (mv^2) / r
Rearrange the equation to solve for the speed (v):
v^2 = (Tension in the upper string * r) / m
v^2 = (48.0 N * 3.87 m) / 1.42 kg
Now, calculate the square root of both sides of the equation to find the speed:
v = √[(48.0 N * 3.87 m) / 1.42 kg]
Solve this expression to find the speed of the ball.