You release a ball from rest attached to a string as shown in Figure A. The ball swings freely, and at the bottom of its circular path it strikes a stationary block. The block slides to the left with a speed of 4ms, and the ball bounces back to the right. To what maximum height does the ball swing back as shown in Figure B? Assume no mechanical energy is lost at any time. (The string’s length is 7 m. The ball’s mass is 1kg, the block’s mass is 4kg, ignore the mass of the string.)
If needed, g=10ms2.
Maximum Height in Figure B =- unanswered meters
2 m
To find the maximum height to which the ball swings back in Figure B, we can use the principle of conservation of mechanical energy. Since no mechanical energy is lost in the system, the total mechanical energy at any point remains constant.
The total mechanical energy of the system is the sum of the kinetic energy and the potential energy. At the top of the swing (Figure B), the ball's kinetic energy is zero since it has reached its maximum height and momentarily comes to rest.
The potential energy at the top of the swing can be calculated using the following formula:
Potential Energy = Mass * Gravity * Height
For the ball, the potential energy at the top of the swing is given by:
Potential Energy (ball) = Mass (ball) * Gravity * Height
We can set up an equation that equates the initial total mechanical energy (just before the collision) to the final total mechanical energy (at the maximum height):
Initial Total Mechanical Energy = Final Total Mechanical Energy
Initial kinetic energy (before the collision) + Initial potential energy (before the collision)
= Final kinetic energy (just after the collision) + Final potential energy (at the maximum height)
Before the collision, the ball is at the lowest point of its swing, where it has only kinetic energy. The equation becomes:
0 + Mass (ball) * Gravity * Height (lowest point)
= 0.5 * Mass (ball) * Velocity^2 (just after the collision) + Mass (ball) * Gravity * Height (maximum height)
Since the ball loses its kinetic energy during the collision and converts it into potential energy at the maximum height, we can rewrite the equation as:
Mass (ball) * Gravity * Height (lowest point)
= Mass (ball) * Gravity * Height (maximum height)
We can now solve for the maximum height:
Height (maximum height) = Height (lowest point)
Given that the length of the string is 7 meters and the ball's initial position is at rest, the lowest point of the swing is reached when the string is fully extended, which is the length of the string:
Height (lowest point) = Length of string = 7 meters
Therefore,
Maximum Height in Figure B = Height (maximum height) = 7 meters