Question

A horizontal disc rotates with a constant angular velocity $\omega$ about a vertical axis passing through its centre. A small body $m$ moves along a diameter with a velocity $v$. Find the force the disc exerts on the body when it is at a distance $r$ located from the rotation axis.

• A. $mr{\omega }^2+2mv\omega$
• B. $mg+\sqrt{m^2{r}^2{\omega }^4+{\left(2mv\omega \right)}^2}$
• C. $\sqrt{m^2{g}^2+{\left(2mv\omega \right)}^2}+mr{\omega }^2$
• D. $m\sqrt{g^2+r^2{\omega }^4+{\left(2v\omega \right)}^2}$

Answer 1

The correct option is D $m\sqrt{g^2+r^2{\omega }^4+{\left(2v\omega \right)}^2}$

Here three force will act on the disc.
1.Vertical gravitational force $=mg$
2. Lateral force perpendicular to vertical plane $=2mvw$
3. The centrifugal force outward from the center $=m{w}^{2}r$
The magnitude of the resultant force $=\sqrt{(mg{)}^2+(2mvw{)}^2+(m{w}^{2}r{)}^2}=m\sqrt{g^2+(2vw{)}^2+w^4{r}^2}$