Q.A bob of mass $m$ is attached to one end of a rod of negligible mass and length $r$, the other end of which is pivoted freely at a fixed centre $O$. [Figure: a rigid, massless rod of length $r$ pivoted at a fixed point $O$, with a bob of mass $m$ at its free end, swinging in a vertical circle.] What initial speed must be given to the object to reach the top of the circle? (Hint: Use the law of conservation of energy.) Is this speed less or greater than the speed obtained in section 4.2.9?

Step 1. Unlike a string, a rigid rod can supply a compressive (pushing) force as well as a tensile (pulling) one. So the rod does not need to maintain a positive tension at the top the way a string does -- the only physical requirement at the top of the loop is that the speed there is not negative, i.e. $v_{\text{top}}\ge0$.

Step 2. The minimum condition is therefore simply $v_{\text{top}}=0$ (the bob just barely reaches the top with zero speed; the rod supplies whatever force -- push or pull -- is needed to keep it on the circular path).

Step 3. Apply energy conservation between the bottom (initial speed $v_{\min}$, taking $U=0$ there) and the top (height $2r$ above the bottom, speed $0$): $\tfrac12mv_{\min}^2 = \tfrac12m(0)^2 + mg(2r)$.

Step 4. Solving: $v_{\min}^2=4gr \Rightarrow v_{\min}=\sqrt{4gr}=2\sqrt{gr}$. …

Recognise that a rigid rod only requires v(top) >= 0 (unlike a string's T >= 0 condition), then apply en …

• Applying the string's condition (v(top) = sqrt(gr)) to a rod problem by mistake, giving the wrong (larger) answer sqrt(5gr). …