(New tips are continually added to these pages. Check back in a few months' time for more)

TOPIC 7: Oscillations and Resonance

Tip 1:

Although not officially in the syllabus, the following equations should be memorised. The alternative, ie. to derive on the spot, may be difficult for the average student.

For a spring-mass system, the natural period of oscillation is given by:

T = 2p Öm/k = 2p Öe/g

where m is the mass of the object attached to the spring; k the spring constant; e the extension of the spring; and g the acceleration due to gravity.

Why is m/k = e/g? That's because mg = ke

For a pendulum system, the natural period of oscillation is given by:

T = 2p Öl/g

where l is the length of the string and g the acceleration due to gravity.

Example 1:

For a spring-mass system, if the the spring is replaced by 2 springs of the same kind in series, and the mass is doubled, what happens to the period of oscillation?

Answer: For 2 springs in series, the spring constant is halved. Since m is doubled and k is halved, m/k will increase by 4 times; and Öm/k will double. Therefore, the period T will double.

Example 2:

If I move a pendulum and a spring-mass system to the moon, where the acceleration due to gravity is 1/6 that on Earth, what happens to the period in each case?

Answer: the period of the pendulum will increase by Ö6 while that of the spring-mass system will remain unchanged.

Reason: since g decreases by 6 times, l/g increases by 6 times. Therefore, Öl/g increases by Ö6.

But what about the spring-mass system? Well, in this case, the mass m is constant (mass does not change); the spring constant k is contant. Hence period T = 2p Öm/k is also constant.

Trick Question:

But for a spring-mass system, isn't period T also = 2p Öe/g ? So shouldn't the period also be affected by g?

Answer: on the moon, g decreases by 6 times, but the extension of the spring also decreases by 6 times since the mass now weighs 6 times less. So the 6 in the numerator cancels out with the 6 in the denominator, and the period remains unchanged.