In the figure, a 120 g ball on a 64.7 cm rope is swung in a vertical circle about a point 299 cm above the ground. If the ball is swung at the slowest speed where the ball goes over the top of the circle without the rope having any slack and the rope is released when the ball is at the top of the loop, how far to the right does the ball hit the floor?
To determine how far to the right the ball hits the floor, we need to analyze the forces acting on the ball and calculate the horizontal displacement. Here's how you can solve the problem step by step:
1. First, let's consider the forces acting on the ball when it's at the top of the loop. At the top, the ball is moving in a circular path, so there is a centripetal force acting towards the center of the circle, provided by the tension in the rope. Additionally, there is the force of gravity acting downward.
2. To find the minimum speed required for the ball to go over the top of the circle without any slack in the rope, we can use the concept of tension and centripetal force. At the top, the tension in the rope must be equal to the sum of the centripetal force and the weight of the ball.
3. The centripetal force is given by the equation Fc = mv^2/r, where Fc is the centripetal force, m is the mass of the ball, v is the velocity of the ball, and r is the radius of the circle (in this case, the length of the rope).
4. The weight of the ball is given by the equation Fg = mg, where Fg is the weight, m is the mass of the ball, and g is the acceleration due to gravity.
5. Since the tension in the rope is equal to the centripetal force plus the weight, we can write the equation as T = Fc + Fg.
6. Solving for v using the above equations, we get v = sqrt((g * r) + g).
7. Now that we have the minimum speed required, we can calculate the time it takes for the ball to reach the ground. Using the equation of motion, h = (1/2)gt^2, where h is the height, g is the acceleration due to gravity, and t is the time, we can solve for t.
8. Given that the ball is released at the top of the loop, the initial vertical velocity is zero. Using v = u + gt, where u is the initial velocity, v is the final velocity, g is the acceleration due to gravity, and t is the time, we can solve for v.
9. Now that we know the final vertical velocity of the ball just before it hits the ground, we can use the horizontal motion of the ball to calculate the horizontal displacement. Since no horizontal forces act on the ball, it will maintain a constant horizontal velocity.
10. The horizontal displacement (distance to the right) can be determined using the equation d = v * t, where d is the horizontal displacement, v is the horizontal velocity, and t is the time calculated in step 7.
By following these steps, you can calculate the horizontal displacement of the ball when it hits the floor.