Final answer:
To find the time required for the ball to stop slipping on the floor, we need to consider the frictional force and torque acting on the ball. By applying the equations for frictional force, rotational inertia, and angular acceleration, we can calculate the time. Once we have the time, we can use the equations of motion to find the distance to the point where the ball is rolling without slipping.
Step-by-step explanation:
To determine the time required for the bowling ball to come to the point where it is not slipping, we need to consider the forces acting on the ball. The frictional force acting on the ball opposes its motion and causes it to slow down. The torque due to the frictional force causes the ball to rotate counterclockwise and eventually stop slipping. The equation that relates the frictional force to the rotational inertia and the angular acceleration is given by:
f = μN = Iα
where f is the frictional force, μ is the coefficient of kinetic friction, N is the normal force, I is the rotational inertia, and α is the angular acceleration.
- Given that the radius of the bowling ball is 8.5 cm, the rotational inertia I can be calculated as I = 2/5 * m * r^2, where m is the mass of the ball.
- The normal force N is equal to the weight of the ball, which can be calculated as N = m * g, where g is the acceleration due to gravity.
- The angular acceleration α can be calculated as the ratio of the frictional force to the rotational inertia: α = f / I.
Using the equation D = ½ α t^2, where D is the distance traveled and t is the time, we can solve for t. Once we have the time, we can use the equation D = v0t + ½ at^2, where v0 is the initial speed and a is the linear acceleration, to calculate the distance D to the point where the ball is rolling without slipping.
Solving these equations will give us the time required for the ball to come to the point where it is not slipping and the distance to that point.