Question

A rotating disc (shown above) moves in the positive direction of the x axis. Find the equation $y\left(x\right)$ describing the position of the instantaneous axis of rotation, if at the initial moment the axis C of the disc was located at the point $O$ after which it moved with a constant velocity $v$, while the disc started rotating counterclockwise with a constant angular acceleration $\beta$ (the initial angular velocity is equal to zero),

• A. $y\left(x\right)=\frac{v^2}{\beta x}$
• B. $y\left(x\right)=3\frac{v^2}{\beta x}$
• C. $y\left(x\right)=\frac{v^3}{\beta x}$
• D. $y\left(x\right)=3\frac{v^3}{\beta x}$

Answer 1

The correct option is A $y\left(x\right)=\frac{v^2}{\beta x}$

The instantaneous velocity of the axis of rotation is zero with respect to outside frame
Here, $y=R$
For disc, moving along $x-$axis, $v=\frac{x}{t}ort=\frac{x}{v}$ and for rotating $w=\frac{v}{R}=\frac{v}{y}$ but, also $w=\beta t$
so, $\frac{v}{y}=\beta tor\frac{v}{y}=\beta \frac{x}{v}or\frac{v^2}{\beta x}=y$