Conservative force and pendulum energy

(a) Conservative force and potential energy

A conservative force is a force whose work between two points depends only on the end positions and not on the path followed (for example gravity and spring force). Work done around any closed path is zero. Friction is non-conservative.

Potential energy $V$ is the energy a body has because of its position or configuration. The work of a conservative force equals the decrease in potential energy: $U_{1\to2} = V_1 - V_2$.

• Gravitational: $V_g = mgh$ (h measured from a chosen datum)
• Elastic (spring of stiffness $k$, extension $x$): $V_e = \tfrac12 kx^2$

Conservation of mechanical energy: if only conservative forces do work, the total mechanical energy remains constant:

$$T_1 + V_1 = T_2 + V_2$$

(b) Pendulum bob

Take the lowest point as datum. Tension does no work (it is perpendicular to the path), so energy is conserved.

   O ____________ (start, v = 0)
   |  \
   |   \ L = 1.2 m
   |    \
   v     o   <- lowest point

$$mgL = \tfrac12 m v^2 \;\Rightarrow\; v = \sqrt{2gL} = \sqrt{2 \times 9.81 \times 1.2} = 4.85\ \text{m/s}$$

At the lowest point the net upward force provides the centripetal force:

$$T - mg = \frac{mv^2}{L}$$ $$T = 0.5 \times 9.81 + \frac{0.5 \times (4.852)^2}{1.2} = 4.905 + 9.810 = 14.71\ \text{N}$$

This equals $3mg$, a known result for release from the horizontal.

Answer: v = 4.85 m/s; T = 14.71 N (= 3mg).