Question

A small ring of uniform density rolls up a curved path with an initial velocity $5{\text{ms}}^{-1}$. Find the maximum height reached by the ring ?

• A. $3\text{m}$
• B. $2.5\text{m}$
• C. $2\text{m}$
• D. $1.5\text{m}$

Answer 1

The correct option is B $2.5\text{m}$


Since there is no non-conservative force acting on the system, thus mechanical energy of the system is conserved.

$E_i=E_f$

Taking ground as a datum line, gravitational potential energy at ground is zero.

$\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2=mg{H}_{max}$

Moment of inertia of the ring, $I=m{R}^2$

$\frac{1}{2}m{v}^2+\frac{1}{2}m{R}^2{\omega }^2=mg{H}_{max}$

Condition for pure rolling,

$v=R\omega \Rightarrow \omega =\frac{v}{R}$

$\frac{1}{2}m{v}^2+\frac{1}{2}m{v}^2=mg{H}_{max}$

$\Rightarrow v^2=g{H}_{max}$

$\Rightarrow H_{max}=\frac{v^2}{g}=\frac{(5{)}^2}{10}=2.5\text{m}$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(B\right)$ is the correct answer.

Why this Question ? This question involves concept of rotation and translational motion. Conservation of energy is most often used law to solve velocity and height.