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A baseball of mass m1 = 0.46 kg is thrown at a concrete block m2 = 8.25 kg. The block has a coefficient of static friction of μs = 0.81 between it and the floor. The ball is in contact with the block for t = 0.165 s while it collides elastically.
m1 = 0.46 kg
m2 = 8.25 kg
μs = 0.81
t = 0.165 s
Part (a) Write an expression for the minimum velocity the ball must have, vmin, to make the block move.
Part (b) What is the velocity in m/s?
To solve this problem, we need to consider the forces acting on the block and the ball during the collision.
Let's start with Part (a) and find the expression for the minimum velocity the ball must have to make the block move.
Part (a):
1. Identify the forces acting on the block:
- The weight of the block (mg2), directed downward.
- The contact force between the block and the floor. This force opposes the impending motion of the block and depends on the coefficient of static friction (μs).
2. The minimum velocity the ball must have to make the block move is when the frictional force exactly balances the weight of the block.
3. Set up an equation for the frictional force:
Since the block is just about to move, the maximum static frictional force is given by: fs_max = μs * (Normal force on the block)
The normal force on the block is equal to the weight of the block (mg2), so the frictional force becomes: fs_max = μs * (mg2)
4. As the ball collides with the block, it exerts a force on the block. This force must be greater than or equal to the maximum static frictional force to induce motion in the block. The force exerted by the ball is given by: force_ball = (Change in momentum of the ball) / (Time of collision)
5. The change in momentum of the ball is equal to its initial momentum, since the collision is assumed to be elastic. The initial momentum of the ball is given by: momentum_ball = m1 * vmin
6. Equate the force exerted by the ball to the maximum static frictional force:
fs_max = m1 * vmin / t
7. Rearrange the equation to solve for the minimum velocity (vmin):
vmin = (fs_max * t) / m1
Now let's move on to Part (b) and calculate the velocity in m/s:
1. Substitute the given values into the equation for vmin:
vmin = (0.81 * 8.25 * 9.8 * 0.165) / 0.46
2. Calculate the value of vmin using a calculator:
vmin = 8.9608 m/s
Therefore, the minimum velocity the ball must have to make the block move is 8.9608 m/s.