Question

# A body initially at rest and sliding along a frictionless track from a height $h$ just completes a vertical circle of diameter $AB=D$. What is the height h equal to?

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Solution

## Step1: Given dataA body initially at rest and sliding along a frictionless track from a height $h$A vertical circle of diameter $AB=D$. Step2: Formula usedPotential energy, $P.E.=mgh$Kinetic energy, $K.E.=\frac{1}{2}m{v}^{2}$Step3: Calculating the heightThe energy at the top= The energy at the bottomAt the top, there is only potential energy and at the bottom, the energy is completely kinetic, so,$mgh=\frac{1}{2}m{v}^{2}\phantom{\rule{0ex}{0ex}}h=\frac{{v}^{2}}{2g}$For completing the vertical circle, the velocity at the lowest point must satisfy the condition,$v\ge \sqrt{5gr}$ where $r$ is the radius and equal to half of the diameter.Now, the value of h becomes,$h=\frac{5gr}{2g}\phantom{\rule{0ex}{0ex}}h=\frac{5}{4}D$Hence, the height h is equal to $\frac{5}{4}D$.

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