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 find the maximum change in length of the spring, we need to use the principles of conservation of mechanical energy.
First, let's calculate the initial kinetic energy of the rolling ball. The equation for the kinetic energy of a rolling object is given by:
KE = (1/2) * I * ω^2 + (1/2) * m * v^2
where KE is the kinetic energy, I is the moment of inertia, ω is the angular velocity, m is the mass of the ball, and v is the linear velocity. Since the ball is rolling without slipping, we can relate the angular velocity to the linear velocity using the equation:
v = ω * r
where r is the radius of the ball.
In this case, the ball is rolling horizontally, so its angular velocity is zero. Therefore, the initial kinetic energy of the ball is simply given by:
KE_initial = (1/2) * m * v^2
where m is the mass of the ball and v is the linear velocity.
Substituting the given values, we can calculate the initial kinetic energy:
KE_initial = (1/2) * 5.3 kg * (4.3 m/s)^2
KE_initial = 48.407 J
At the maximum change in length of the spring, the ball will come to a momentary stop, meaning its final kinetic energy will be zero. We can equate the initial kinetic energy to the potential energy stored in the spring when it reaches its maximum compression. The equation for the potential energy of a mass attached to a spring is given by:
PE = (1/2) * k * x^2
where PE is the potential energy, k is the spring constant, and x is the change in length of the spring from its equilibrium position.
In this case, we want to find the maximum change in length, so we need to solve the equation for x:
x = √(2 * KE_initial / k)
Substituting the given values, we can calculate the maximum change in length of the spring:
x = √(2 * 48.407 J / 41100 N/m)
x ≈ 0.258 m
Therefore, the maximum change in length of the spring is approximately 0.258 meters.