Skidding and rolling without slipping of a bowling ball

[chenying]

Homework Statement

A bowler throws a bowling ball of radius R = 11 cm down the lane with initial speed v0 = 8.5 m/s. The ball is thrown in such a way that it skids for a certain distance before it starts to roll. It is not rotating at all when it first hits the lane, its motion being pure translation. The coefficient of kinetic friction between the ball and the lane is 0.22.

(a) For what length of time does the ball skid? (Hint: As the ball skids, its speed v decreases and its angular speed ω increases; skidding ceases when v = Rω.)

(b) How far down the lane does it skid?

(c) How fast is it moving when it starts to roll?

Homework Equations

v=r$$\omega$$

$$\omega$$f = $$\omega$$i - $$\alpha$$t

$$\tau$$ = I$$\alpha$$

The Attempt at a Solution

Ok...I really have no idea where to start. The clue they gave me gives me some ideas, but I still need some clarification. When the bowling ball starts to skid, does it have an initial angular speed? I know there is an initial and final velocity for the ball, but I'm confused about the angular speed of the ball.

Please helpppp

 

[ehild]

The motion of a ball consist of the translation of its centre of mass and rotation around the centre of mass. When it rolls, the displacement of the CM during one rotation is equal to the circumference, $s=r \omega$. (You can see it on a roll of paper), that is why $v=r \omega$ when the ball only rolls and do not skids.

When skidding, force of kinetic friction acts at the bottom where the ball touches the ground. This force decelerates the translational motion but its torque accelerates rotation.

Write the equation both for acceleration of CM and angular acceleration. At the beginning, the ball only skids, that is the angular velocity is 0. Determine how both the velocity of the CM and angular velocity of rotation depend on time. Find the time when $v=r \omega$ .

ehild

 

[kerimek]

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I would suggest starting by breaking down the problem into smaller parts and identifying the relevant equations and concepts that can be applied.

First, we can consider the motion of the bowling ball as it skids on the lane. This can be described using the equations of kinematics, specifically the equation for displacement in terms of initial velocity, acceleration, and time (x = v0t + 1/2at^2). The acceleration in this case is due to the friction between the ball and the lane, which can be calculated using the coefficient of kinetic friction and the weight of the ball.

Next, we can consider the rotation of the ball as it skids. The ball is initially rotating purely in translation, but as it skids, the friction force will cause it to start rotating. This can be described using the equations for rotational motion, such as the relationship between linear velocity and angular velocity (v = rω) and the equation for torque (τ = Iα). The moment of inertia, I, for a solid sphere can be calculated using its mass and radius.

Using these equations, we can determine the length of time the ball skids (a), the distance it travels while skidding (b), and the final velocity of the ball when it starts to roll (c). It may also be helpful to draw a free body diagram to visualize the forces acting on the ball and how they change as it skids and starts to roll.

In conclusion, the problem can be solved by applying the laws of motion and rotational motion, using the given information and relevant equations. It is important to break down the problem into smaller parts and to consider the different aspects of the ball's motion (translation and rotation) separately.