**(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 Ö(*m/k*) = 2p Ö(*e/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 Ö(*l/g*).

where *l* is the length of the string
and *g* the acceleration of free fall.

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 quadruple; and Ö(*m/k*) will double. Therefore, the period
*T* will double.

Example 2:

If I relocate both 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 be Ö6 times that on Earth, while that of the spring-mass system will remain unchanged.

Reason: since *g*
becomes 1/6
times its original value, the ratio *l/g* becomes 6 times its original
value. Therefore, Ö(*l/g*)
becomes Ö6
times its original value.

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 constant. Hence period *T* = 2p Ö(*m/k*) is also constant.

Trick Question:

But for a spring-mass system, isn't
period *T* also = 2p Ö(*e/g*) ? So
shouldn't the period also be affected by *g* as well?

Answer: on the moon, *g*
is 1/6 times, but the extension of the spring also becomes 1/6 times since the mass now
weighs only 1/6 of its weight on Earth. So the 1/6 in the numerator cancels out with the
1/6 in the
denominator, and the period remains unchanged.

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