This question is about resonance in vibrating objects.

PHYSICS · Multi-part · A Level · Physics · Further mechanics and thermal physics · Further mechanics and thermal physics

Problem

This question is about resonance in vibrating objects. (1) Write down the period of the function

\cos \pi t

. (2) i Sketch the function

y = \cos \pi t + \cos 2\pi t

for

0 \le t \le 3

. ii Write down the period of the function

\cos \pi t + \cos 2\pi t

. (3) i Use technology to investigate the period of the given functions below. Write down the values of A, B and C.

<b>f(t)</b>	<b>Period</b>
$\cos \pi t + \cos 1.5\pi t$	A
$\cos \pi t + \cos 1.25\pi t$	B
$\cos \pi t + \cos 1.1\pi t$	C

ii Hence conjecture an expression for the period,

T

, of

f(t) = \cos \pi t + \cos \left(1 + \frac{1}{n}\right)\pi t

where

n

is an integer. (4) Prove that, for your conjectured value of

T

f(t + T) = f(t)

. (5) i Use the compound angle formula to write down and simplify an expression for

\cos(A + B) + \cos(A - B)

. ii Hence find a factorized form for the expression

\cos P + \cos Q

. (6) By considering the factorized form of

f(t)

, explain the shape of its graph. (7) A piano string oscillates when plucked. The displacement,

x

, from equilibrium as a function of time is modelled by:

\frac{d^2x}{dt^2} + 4x = 0.

Show that a function of the form

x = f(t) = \cos(\omega t)

solves this differential equation for a positive value of

\omega

to be stated. (8) The piano string can be subjected to an external driving force from a tuning fork. The differential equation becomes:

\frac{d^2x}{dt^2} + 4x = \cos kt.

Find a solution of the form

x = f(t) + g(k) \cos kt

where

g(k)

is a function to be found. (9) Resonance is a phenomenon in which the amplitude of the driven oscillation grows without limit. For what positive value of

k

will resonance occur? Justify your answer.

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