A bowling ball weighing 71.2 is attached to the ceiling by a 3.50 rope. The ball is pulled to one side and released; it then swings back and forth like a pendulum. As the rope swings through its lowest point, the speed of the bowling ball is measured at 4.20. 1)At that instant, find the magnitude of the acceleration of the bowling ball.2)At that instant, find the direction of the acceleration of the bowling ball. 3) At that instant, find the tension in the rope.
mass of ball = w/g
Acentripetal = v^2/R
Force in rope = T up
force of gravity down = m g = 71.2 N
F = m a
T -W = Ac*M
T = W + Ac*M
To answer these questions, we can use the principles of circular motion and Newton's laws.
1) To find the magnitude of the acceleration of the bowling ball at the lowest point, we need to consider the forces acting on it. At the lowest point, the ball has a maximum speed but zero vertical component of velocity. This means that the net force on the ball is only directed towards the centripetal direction, which causes the ball to move in a circular path.
The net centripetal force acting on the ball is provided by the tension in the rope. At the lowest point, the tension force is responsible for the centripetal acceleration. So we can write:
Tension force = Mass x Centripetal acceleration
Since the ball is at the lowest point, its weight (mg) acts directly downwards. Therefore, the tension in the rope is equal to the weight of the ball:
Tension = Weight = mass x gravity
Here, the mass of the ball is given as 71.2 kg, and the acceleration due to gravity is typically 9.8 m/s^2.
So the magnitude of the acceleration at the lowest point is:
a = g = 9.8 m/s^2.
2) The direction of the acceleration can be determined by considering the motion of the ball at the lowest point. The acceleration is directed towards the center of the circular path, i.e., it is directed upwards in this case. So the direction of the acceleration is opposite to the direction of the ball's velocity at the lowest point.
3) The tension in the rope can be found using the equation Tension = Mass x Acceleration. At the lowest point, the acceleration is equal to the gravitational acceleration (9.8 m/s^2) and the mass is given as 71.2 kg.
So the tension in the rope at the lowest point is:
Tension = 71.2 kg x 9.8 m/s^2 = 698.56 N.
Therefore, at the instant when the ball reaches its lowest point, the magnitude of the acceleration is 9.8 m/s^2 directed upwards, and the tension in the rope is 698.56 N.