A ball of mass 5.3 kg and radius 8 cm rolls without slipping horizontally at 4.3 m/s and hits a spring attached to a wall. What is the maximum change in length of the spring if the spring constant k=41100N/m? (Assume the floor is frictionless)
To determine the maximum change in length of the spring, we need to consider the conservation of mechanical energy.
The ball starts with kinetic energy due to its linear motion, and when it hits the spring, the energy is converted into potential energy stored in the compressed spring.
The kinetic energy of the ball can be calculated using the formula:
KE = (1/2) * m * v^2
where m is the mass of the ball and v is its velocity.
KE = (1/2) * 5.3 kg * (4.3 m/s)^2
= 49.7965 J
The potential energy stored in the compressed spring can be calculated using the formula:
PE = (1/2) * k * x^2
where k is the spring constant and x is the change in length of the spring.
To find the maximum change in length, we equate the kinetic energy to the potential energy:
KE = PE
(1/2) * m * v^2 = (1/2) * k * x^2
Rearranging for x, we get:
x = sqrt( (m * v^2) / k )
x = sqrt( (5.3 kg * (4.3 m/s)^2) / 41100 N/m )
x = sqrt( (5.3 kg * 18.49 m^2/s^2) / 41100 N/m )
x = sqrt( 96.997 kg * m^2/s^2 / 41100 N/m )
x = sqrt( 2.360 * 10^-3 m^4/kg/s^2 )
x = 0.048583 m
Therefore, the maximum change in length of the spring is 0.0486 m, or approximately 4.86 cm.