A ferris wheel turns at a constant speed. the lowest point of the wheel is ground level. a person is standing on a platform 4m about ground. The intervals between two successive times are alternatly 6 s and 18 s. Part A) what is the period of rotation? Part B) what is the raduis of the wheel? Part C) graph the function?
A) 18-6 = 12 seconds
B) no idea of the radius. You give no indication of the height of the wheel, or where the platform is in relation to the axle.
C) no can do. All we know is that
h(t) = A + Bsin(pi/6 t + C)
Part A) To find the period of rotation, we need to determine the time it takes for the ferris wheel to complete one full revolution. We are given that the intervals between two successive times are alternately 6 seconds and 18 seconds. Let's analyze the pattern:
- At the first time interval of 6 seconds, the person is at the same position as the starting point. This can be considered as completing 0.5 revolutions.
- At the second time interval of 18 seconds, the person is again at the starting point, which indicates a full revolution.
So, in a total time of 6 + 18 = 24 seconds, the ferris wheel completes 1.5 revolutions. Therefore, we can determine the period of rotation as:
Period = Total Time / Number of Revolutions
= 24 seconds / 1.5 revolutions
= 16 seconds
Therefore, the period of rotation for the ferris wheel is 16 seconds.
Part B) To find the radius of the wheel, we need to make use of the formula for linear velocity (v) in uniform circular motion:
v = 2πr / T
Where v is the linear velocity, r is the radius, and T is the period of rotation. We know that the linear velocity remains constant throughout the motion.
Since the person on the platform is standing 4 meters above the ground, the distance traveled will be the circumference at ground level (2πr) plus the 4 meters. In one complete revolution, this distance will be covered in 18 seconds (the time interval for a complete revolution).
Distance = 2πr + 4 meters
Time = 18 seconds
We can rearrange the formula for linear velocity to solve for the radius (r):
v = 2πr / T
2πr / 18 = v
Since we are given that the intervals alternate between 6 seconds and 18 seconds, the linear velocity remains constant. So we can say:
2πr / 6 = 2πr / 18
18 * 2πr = 6 * 2πr
r = 6 meters
Therefore, the radius of the ferris wheel is 6 meters.
Part C) To graph the function, we need to represent the height of the person on the platform as a function of time. We can identify two repeating intervals: 6 seconds and 18 seconds.
In the first 6 seconds, the person starts at a height of 4 meters and returns to the same point. Therefore, this segment can be represented by a horizontal line at a height of 4 meters.
In the next 18 seconds, the person completes a full revolution and returns to the same point. Therefore, this segment can be represented by a sinusoidal wave starting from a height of 4 meters, reaching a maximum, and then returning to 4 meters.
Overall, the graph will consist of alternating horizontal lines at 4 meters and sinusoidal waves.