More Examples

1. A mass $m$ is hanging by a massless thread of length $L$ and is given an initial speed $v_0$ to the left. It swings up and stops at some maximum height $H$ an angle $\theta$. Find $\theta$.

   We solve this by setting $E_i = E_f$ (total energy is conserved).

   Initial:
   

   $$E_{\rm tot} = \frac{1}{2}m v_0^2 + mg(0) = \frac{1}{2}m v_0^2$$ (3.41)

   
   

   Final:
   

   $$E_{\rm tot} = \frac{1}{2}m (0)^2 + mgh = mgL(1 - \cos(\theta))$$ (3.42)

   
   

   Set them equal and solve:
   

   $$\cos(\theta) = 1 - \frac{v_0^2}{2gL}$$ (3.43)

   
   
   or
   

   $$\theta = \cos^{-1}(1 - \frac{v_0^2}{2gL}).$$ (3.44)

   
   

2. Mass pushed by compressed spring that slides up smooth incline to height $H$? Show that $H = \frac{k\Delta x^2}{2mg}$.

3. Mass on rough incline of height $H$ (given) slides down to bottom. Find $v_f$ at the bottom? (Set $E_f - E_i = W_f$, where the latter is negative, and solve for $v_f$).

4. Loop the loop. What is the minimum height $H$ such that the mass m loops-the-loop (stays on the track around the circle). Ans: $H = 5/2 R$.

5. Block and tackle. Tension $T$ doesn't do any net work on system, just redistributes energy. Potential energy of gravity is distributed throughout the system as kinetic energy.