Tuesday, December 6, 2016

A small ball rests at point P on a curved track of radius r ...







Question 1
(a) State what is meant by simple harmonic motion. [2]

(b) A small ball rests at point P on a curved track of radius r, as shown in Fig. 4.1.


Fig. 4.1
The ball is moved a small distance to one side and is then released. The horizontal displacement x of the ball is related to its acceleration a towards P by the expression
a = − gx / r
where g is the acceleration of free fall.

(i) Show that the ball undergoes simple harmonic motion. [2]

(ii) The radius r of curvature of the track is 28 cm.
Determine the time interval τ between the ball passing point P and then returning to point P. [3]

(c) The variation with time t of the displacement x of the ball in (b) is shown in Fig. 4.2.


Fig. 4.2
Some moisture now forms on the track, causing the ball to come to rest after approximately 15 oscillations.
On the axes of Fig. 4.2, sketch the variation with time t of the displacement x of the ball for the first two periods after the moisture has formed. Assume the moisture forms at time t = 0. [3]



Reference: Past Exam Paper – November 2014 Paper 41 & 42 Q4

 

Solution 1:
(a) For a simple harmonic motion, the acceleration / force is proportional to the displacement (from a fixed point) and the
EITHER acceleration and displacement are in opposite directions                OR acceleration is always directed towards a fixed point

(b)
(i) Acceleration due to gravity, g and radius r are constant, so the acceleration a is proportional to the displacement x.
The negative sign shows that a and x are in opposite directions.

(ii)
{For simple harmonic motion, acceleration a = – ω2x
a = – ω2x = – gx / r giving}
ω2 = g / r                      and      ω = 2π / T
ω2 = 9.8 / 0.28 = 35                
T = {2π / ω2 =} 2π / √35 = 1.06 s
{The ‘T’ calculated above is the period. The question asks for only the time interval τ between the ball passing point P and then returning to point P. This is half the period T.}
time interval τ = {T/2 =} 0.53 s

(c)
sketch: time period constant (or increases very slightly)
drawn line always ‘inside’ given loops
successive decrease in peak height



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