Question

The maximum speed of the pendulum bob in a grandfather clock is 0.55m/s .

If the pendulum makes a maximum angle of 5.4? with the vertical, what is the length of the pendulum?

Answer 1

Treat the pendulum bob as a point mass and neglect the mass of the rod.

Express the kinetic energy of the bob at lowest point:
KE_max = 1/2*m*v_max^2

Express the potential energy of the bob at highest point:
GPE_max = m*g*h_max

The height above the lowest point is given by the following. Draw a triangle to visualize its origin.
h = L*(1-cos(theta))

Modify GPE equation:
GPE_max = m*g*L*(1-cos(theta_max))

Assuming a conservative pendulum, GPE_max = KE_max

Therefore:
m*g*L*(1-cos(theta_max)) = 1/2*m*v_max^2

Divide through by m and solve for L:
L = v_max^2/(2*g*(1 - cos(theta_max)))
L= 3.47