A Ferris wheel with a radius of 9.61 m rotates at a constant rate, completing one revolution every 34.1 s. Calculate the magnitude of a passenger's acceleraton when at the top of the wheel.
Centripetal acceleration, ac
= rω²
The direction is radially outwards
At the bottom, radial acceleration (measured positive upwards) adds to gravity
= -ac-g
At the top of the wheel, radial acceleration (measured positive upwards) reduces gravity
= ac-g
To calculate the magnitude of a passenger's acceleration when at the top of the Ferris wheel, you can use the following steps:
Step 1: Find the angular velocity:
The angular velocity (ω) is the rate at which the Ferris wheel completes one revolution. It can be calculated by dividing 2π (the number of radians in a circle) by the time taken for one revolution (T). In this case, T is given as 34.1 seconds.
ω = (2π) / T
ω = (2π) / 34.1
Step 2: Calculate the linear velocity:
The linear velocity (v) can be calculated by multiplying the angular velocity (ω) by the radius (r) of the Ferris wheel.
v = ω * r
v = ω * 9.61
Step 3: Determine the centripetal acceleration:
The centripetal acceleration (a) is the acceleration experienced by an object moving in a circular path. It can be calculated using the formula:
a = v^2 / r
a = (v * v) / r
Step 4: Substitute the values and calculate the magnitude of the passenger's acceleration:
Substitute the calculated values of v and r into the formula for centripetal acceleration (a) to find the magnitude of the passenger's acceleration when at the top of the Ferris wheel.
a = [(ω * r) * (ω * r)] / r
a = ω^2 * r
Finally, substitute the calculated values of ω and r into the formula:
a = (2π / 34.1)^2 * 9.61
You can solve this equation to find the numerical value of the magnitude of the passenger's acceleration when at the top of the Ferris wheel.