Q7.12 Swinging stick (H)
A meter stick oscillates in the vertical plane about a pivot point at one end.- Show that the relation between torque and angular acceleration (Key Point 5.19) leads to the SHM equation (Key Point 6.2); identify the angular frequency and deduce the period.
- Determine the length of a simple pendulum which would have a period equal to that of the meter stick.
- Discuss what happens if the pivot point is at the centre of the stick.
Hint
Reveal

Hint
You will need to look up, or work out the appropriate moment of inertia. Then make use of the result of Equation 6.8; but think about it first.Solution
Reveal

Solution
This is a special case of the general treatment of the physical pendulum we discussed in S6.4
In its stable equilibrium position the stick hangs down from the point of suspension.
Suppose it is displaced slightly from this arrangement, so that it makes a small angle θ with the vertical.
We will need the angular form of Newton’s 2nd Law (Key Point 5.19):
To make use of this equation we need the moment of inertia I of the rod about its end.
You can look this up, or perhaps work it out:
We also need the torque τ about the pivot. The torque comes from the forces that act on the stick:
- its weight, mg, acting downwards, through the CoM
- the reaction force FP, exerted by the pivot, acting through the pivot.
The pivot force exerts no torque about the pivot point. The torque exerted by the weight (gravitational force) has the magnitude
The equation of motion follows after some thought about signs
[You will find some help about the signs in the section on the physical pendulum in S6.4.]
Rearranging gives the EOM
Provide the oscillations are small so that we can make use of the TSE sinθ≃θ (TSE means Taylor Series expansion in case you’re just dropping in from another planet) the EOM may be written in the standard form
The period follows as- Now for a simple pendulum of length L0 the period is
Equating T0 and T we find that a simple pendulum will have the same period as the metre stick if its length is:
- If the pivot point were at the centre of the metre stick (rather than the end) an angular displacement generates no restoring torque (and therefore no oscillation) because then the gravitational force would also act through the pivot point. This arrangement is an example of neutral equilibrium.