Question

A mouse wheel is 20cm in diameter and the base of the wheel is 1cm off the ground. If the hamster can spin the wheel 3 times each second, what s the sine function that describes the movement of the wheel of it starts at its lowest point?

Answer 1

Given that a mouse wheel is 20cm in diameter and the base of the wheel is 1cm off the ground. And a hamster can spin the wheel 3 times in each second.

So the wheel undergoes a periodic motion in a circle with time period `T=1/3sec`
i.e.with angular velocity `omega=(2pi)/(1/3)=6pi` rad/s.

The diameter of the wheel being `20cm` any point on the periphery of the wheel changes its height from `1cm` to `21cm` centering about the center of the wheel.So the amplitude of the periodic motion will be `A=10cm`

If we describe the the movement of the lowest point point of the wheel from its start as a cosine function then we get the following function to represent the change in height of its lowest point w .r .t time `t`.

`h(t)=11-Acos(omega*t)`

`=>h(t)=11-10cos(6pi*t)`

To have a clear view of 3 cycles in one second in the graphical representation of the function the variation of time unit is diminished 50 times to `0.02` sec . The function then takes the following form

`=>h(t)=11-10cos((6pi*t)/50)`

Expressing this in sine function as desired we get

`h(t)=11-10sin(pi/2-(6pi*t)/50)`

`color(red)(=>h(t)=11+10sin((6pi*t)/50-pi/2))`