13.1.1 SHM
Simple pendulum
Simple pendulum
A pendulum consists of a small “bob” of mass m, suspended by a light inextensible thread of length l, from a fixed point
We can ignore the mass of the thread
The bob can be made to oscillate about point O in a vertical plane along the arc of a circle
We can show that oscillating simple pendulums exhibit SHM
We need to show that a x Consider the forces acting on the
pendulum: weight, W of the bob and the tension, T in the thread
We can resolve W into 2 components parallel and perpendicular to the thread:
1. Parallel: the forces are in equilibrium2. Perpendicular: only one force acts,
providing acceleration back towards O
Parallel: F = mg cos Perpendicular: F = restoring force towards O
= mg sin This is the accelerating force towards O F = ma - mg sin = ma (-ve since towards O)
When is small (>10°) sin Hence -mg = ma (remember = s/r = x/l)
-mg (x/l) = ma Rearranging: a = -g (x/l) = pendulum equation
(can also write this equation as a = -x (g/l))
In SHM a x Since g/l = constant we can assume a x for
small angles only SHM equation a = -(2f)2x Pendulum equation a = -x (g/l) Hence (2f)2 = (g/l) f = 1/2 (g/l) remember T = 1/f T = 2 (l/g) The time period of a simple pendulum
depends on length of thread and acceleration due to gravity
Measure acceleration of free fall using simple pendulums
Use page 36 and 37 of “Physics by Experiment”
Set up the equipment and set the length of the string so T = 2s
Mark a reference point on the stand (to count number of oscillations
Displace the pendulum a few centimetres and release – the swing should be 1 plane
As the pendulum passes the reference point start the stopwatch and measure the time for 20 oscillations
Remember 1 oscillation is from OA O B O
Now change the length of the string (shorter or longer), measuring the length from the point of suspension to the centre of gravity of the bob
Repeat the experiment Record results in a table with the column
headings T, T2 and y (in metres) Plot a graph T2 against y – this should be a
straight line graph
Analysis of results
T = 2 (l/g) T2 = 42 (l/g) g = 42 (l/T2) From your graph find a value for g (from gradient) 1. How does your value of g compare to the
accepted value?2. This experiment requires that (i) 20 oscillations be
timed (ii) the angle of the swing is small (iii) the length of the string is measured to the centre of the bob and (iv) the oscillations are counted as the bob passes the equilibrium point. Why?
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