The drawing shows a golf ball passing through a windmill at a miniature golf course. The windmill has 12 blades and rotates at an angular speed of 1.35 rad/s. The opening between successive blades is equal to the width of a blade. A golf ball (diameter 4.50 x 10-2 m) has just reached the edge of one of the rotating blades. Ignoring the thickness of the blades, we need to find the minimum linear speed with which the ball moves along the ground so that it will not be hit by the next blade.
Answer:
v_min = 0.23 m/s
Explanation:
The golf ball must travel a distance equal to its diameter in the time between blade arrivals to avoid being hit. With 12 blades and 12 blade openings of the same width, each blade or opening is 1/24 of a circle or 2π/24 = 0.26 radians across.
Therefore, the time between the edge of one blade moving out of the way and the next blade moving in the way is given by:
time = angular distance / angular velocity
⇒ t = 0.26 rad / 1.35 rad/s = 0.194 s
The golf ball must get completely through the blade path in this time, so it must move a distance equal to its diameter in 0.194 s. Therefore, the speed of the golf ball is:
v = d / t
⇒ v = 0.045 m / 0.194 s = 0.23 m/s
Final Answer:
To calculate the golf ball's minimum linear speed to avoid being hit by the next windmill blade, we derive the time it takes for one blade to pass and then determine the speed that allows the ball to travel its own diameter within that time.
Calculating the Minimum Linear Speed for a Golf Ball
To calculate the minimum linear speed at which the golf ball must move to not be hit by the next blade of the windmill, we have to consider how long it takes for one blade to move out of the way and the next one to enter the same position. Since the windmill has 12 blades and rotates with an angular speed of 1.35 rad/s, the time for one blade to pass and create space for the golf ball is the time it takes for the windmill to rotate the angle equivalent to one blade width plus one opening.
First, we find the time for one complete revolution by taking the ratio of 2π radians to the angular speed:
T = 2π / ω, where ω is the angular speed.
Then we divide T by the number of blades to find the time for one blade to pass:
t = T / 12.
Because the ball's diameter is 4.50 x 10-2 m and it has to pass through an opening of the same size, it needs to clear the space within the time t. Therefore, its minimum speed is the diameter divided by t: v = d / t.
Using these steps, we can find the required linear speed for the golf ball to avoid the next blade.
How is the minimum linear speed of the golf ball calculated to avoid being hit by the next windmill blade?
The minimum linear speed of the golf ball is calculated by determining the time it takes for one blade to pass and creating space for the golf ball. By dividing the ball's diameter by this time, we can find the required speed for the ball to avoid the next blade.