With compound pendulum, the oscillation formula:
J: Moment of inertia of pendulum via oscillation axis (H1 or H2).
s: Distance from weight (m2) to oscillation axis (H1 or H2).
g: Gravitational acceleration.
m: Mass of pendulum.
The reduced length Sr of the compound pendulum is defined as the quantity:
because its oscillation period:
corresponds to that of a simple pendulum with the length Sr.
The moment of inertia J of the compound pendulum is, according to the parallel axis theorem:
Js: moment of inertia around the centre of mass axis.
Therefore the reduced length of pendulum is:
With reversible pendulum:
The reversible pendulum is a particular type of the compound pendulum. There are two edges H1 and H2 that allow to choose the suspension point.. Two weights m1 = 940g và m2 = 1360g can be shifted so that the oscillation period is tunable. The goal of the tuning is to achieve equal oscillation periods around both edges
When T1 = T2, In this case, the reduced length of pendulum is equal:
s1 and s2 is distance from weight m2 to H1 anf H2. Then:
d: distance from H1 to H2.
From formula (6) and (7):
S1 is inserted in Eq (5):
Set up and procedure:
Reversible Pendulum (Kater's pendulum)
- Mount the wall holder on a stable, non-vibrating wall.
- Attach photogate to wall (by screw), so that pendulum scan to IR eye of photogate (see the clip below)
- Fix the mass m1 at the position x1 = 10 cm from H1 (position of m1 can be changed)
- We do experiment with m2 at position x2 = 9.7; 12.2; 14.7 cm; .... 69.7cm from H1 (the top line of pendulum is away from axis of oscillation H1 9.7cm)
Measure period by 2 ways:
+ The first way: Select Mode 4 in photogate timer. To measure period in directly.
+ Second way: Select Mode 5 in photogate timer: Count 50 periods, measure total time. Then calculate average period.
We recommend the second way for better accuracy.
- Deflect the pendulum a small angle, the rod of reversible pendulum cover IR eye of photogate. Release the rod, the Kater's pendulum will oscillate.
- Press Start/Stop button to count. Wait for counting 50 periods then press Start/Stop again to stop count.
- Write the total time for 50 periods to data table.
- Slide the weight m2 to second line position (x2 = 12.2cm). Measure 50 periods again.
- After we do with the 25th line (x2 = 69.7cm). We reverse the pendulum, the axis of oscillation is H2.
Reversible Pendulum: Axis H2
- We do experiment again similar to doing with axis H1.
- We have the experiment data below:
Use this formula to determine the gravitational acceleration:
The graphs T1 and T2 intersect at: 3.154 and 3.334
From formula (10) above, so g1 = 10m/s2, g2 = 9.46m/s2
Average value: g = 9.73m/s2