The figure shows a golf ball passing through a windmill at a miniature golf course. The windmill has 8 blades and rotates at an angular speed of 2.02 rad/s. The opening between successive blades is equal to the width of a blade. A golf ball of diameter 3.27 x 10-2 m is just passing by one of the rotating blades. What must be the minimum speed of the ball so that it will not be hit by the next blade?
I understand the how this works logically just not how to arrive at the answer. thanks
To find the minimum speed of the ball so that it will not be hit by the next blade, we need to consider the time it takes for the ball to clear a blade and the time it takes for the next blade to reach the same position.
First, let's find the time it takes for the ball to clear a single blade.
The golf ball must travel a distance equal to the diameter of the blade to clear it. We are given that the diameter of the golf ball is 3.27 x 10^(-2) m. Since the opening between successive blades is equal to the width of a blade, the distance the ball needs to cover to clear the blade is also 3.27 x 10^(-2) m.
To find the time it takes for the ball to travel this distance, we can use the equation:
time = distance / speed
In this case, the distance is 3.27 x 10^(-2) m.
Now let's calculate the time it takes for the next blade to reach the same position.
The windmill has 8 blades, indicating that the time it takes for one complete revolution of the windmill (360 degrees) is equal to the time it takes for all 8 blades to pass by the initial position.
We are given that the windmill rotates at an angular speed of 2.02 rad/s. The time it takes for the windmill to complete one revolution can be found using the equation:
time = (2π) / angular speed
where π is approximately 3.14.
After finding the time it takes for one revolution, we can divide it by 8 to find the time it takes for a single blade to reach the starting position again.
Now, we can compare the time it takes for the ball to clear a blade and the time it takes for the next blade to reach the same position.
If the time it takes for the ball to clear a blade is less than the time it takes for the next blade to reach the same position, the ball will not be hit by the next blade.
So, the minimum speed of the ball can be calculated using the following steps:
1. Calculate the time it takes for the ball to clear a blade using the equation: time = distance / speed, where distance is the diameter of the ball.
2. Calculate the time it takes for the next blade to reach the same position using the equations: time = (2π) / angular speed for one revolution, and then divide by 8 to get the time for a single blade.
3. Compare the time taken for the ball to clear a blade with the time for the next blade to reach the same position. If the time for the ball is less, the minimum speed is sufficient to clear the blade without being hit.
Note: Make sure to convert units appropriately and use the given values accurately in calculations.