A wheel of radius $4$cm rolls along the x-axis with angular velocity $2$rad/s. Find parametric equations for the curve described by a point on a spoke 2cm from the centre of the wheel if it starts from the point $(0,2)$ at time $t=0$.
Since you assume the wheel is a circle I tried to use a sin a cos function but could not get the answer correct. I tried the equations in the form $x=a\cos(bt)$ and $y=\sin(bt)$. I am also struggling to understand how to incorporate the angular velocity of the wheel into the parametric equations.
Best Answer
Start by looking at the center of the wheel. At time $t$ the center will have position of: $(2t,4)$
As the wheel has moved forward a distance of $2t$ cm in time $t$ then this length of the curve will have rolled along. So the angle turned $\theta$ at time $t$ will be given by: $\theta=\frac{2t}{4}=\frac{t}{2}$.
So any point on the curve has to combine these two motions - the linear motion of the center moving forward and the rotational motion of the point around the center.
The rotational motion around the center is given by $\left(-2\sin\frac{t}{2},-2\cos\frac{t}{2}\right)$ compared to the center of the circle during this time. Note the signs are such as to give the correct starting position and direction of rotation around the center.
Putting these together gives position, $\overline{p}(t)$ as:
$$\overline{p}(t)=(2t,4)+\left(-2\sin\frac{t}{2},-2\cos\frac{t}{2}\right)$$
$$\overline{p}(t)=\left(2t-2\sin\frac{t}{2},4-2\cos\frac{t}{2}\right)$$
This type of curve is know as a Curtate Cycloid.