# S.H.M. with a cantilever

##### Class practical

This experiment could extend (or replace) the traditional pendulum or mass-on-a-spring experiments illustrating S.H.M.

#### Apparatus and materials

Metre rules

G-clamps

Slotted masses (100 g each)

Sellotape

Stopwatches

Small, rough wooden blocks

#### Health & Safety and Technical notes

It might be best advised to wear goggles in case something snaps.

Don't stand with toes underneath the slotted masses.

#### Procedure

a G-clamp the metre rule securely to the bench using the wooden blocks to protect the rule and bench.

b Sellotape one or more slotted masses near the end of the rule.

c Twang and time several oscillations.

d Adjust the vibrating length (or mass attached) and repeat.

#### Teaching notes

1 You could use this experiment as a follow-up to the standard "g from a pendulum". There are more variables to play with so you can easily set up differentiated tasks for your students.

2 In this case we can find E for wood because for this cantilever we have ...
ω2 = Exy3 / 4ML3 where

x = width of ruler
y = thickness of ruler (scale to undersurface)
M = mass Sellotaped on
L = vibrating length
E = Young modulus of wood
ω = 2π/T

So T2 v M (or L3) gives you E from the gradient.

3 With a wide range of abilities you can have one group simply verifying it's S.H.M. (by proving T is independent of amplitude), another determining E, and another using log graphs to discover thatT is proportional to L3/2. It's also a good one for error analysis; which term contributes the largest error in E (answers on a postcard)?

The vibrations are quite fast (especially at short lengths). To obtain an accurate result for T, time many oscillations and find the average time for a single oscillation.

4 If you have the materials you can try things other than metre rules.

Thank you to Wayne Morton for pointing out that there was an error in the formula that we previously printed.

This experiment was submitted by Jason Welch who is Head of Physics at County High School, Leftwich, Cheshire.