Chapter 1   Question 3
 
a. A particle P moves at a constant speed v in a circular path of radius r as shown. 
c0103a.gif (3248 bytes) 

i)    Write down expressions for the time to complete one cycle and the number of cycles performed per unit time.

 12 marks
 
The time for completing one cycle is the time to move through 2pr using speed v. 
 wpe19.jpg (1133 bytes) 
The number of cycles per unit time is the frequency. It is the reciprocal of T. 
 wpe1A.jpg (1373 bytes)		 2
 
ii)     At time t = 0, the particle passes through the origin O. Roughly draw the following graphs for one complete cycle, taking into account the directions of displacement: 
    (1) x-t graph, 
    (2) y-t graph.
 
In chapter 5 (Simple harmonic motion), you will learn that the projection of circular motion on a diameter is simple harmonic motion. SHM could be represented by either 

wpe1B.jpg (2265 bytes) 

The projection of P on x-axis is represented by 

wpe1C.jpg (1456 bytes) 
This is because the motion starts from the leftmost side and at t = 0 , x = 0. 
The motion is graphically represented as shown in Fig.1.3.2.

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c0103b.gif (5623 bytes) 2, 2
The projection of P on the y-axis is represented by 

wpe1E.jpg (1144 bytes) 
This is because the motion starts from origin and moves upwards. At t = 0 , y = 0. 
The motion is graphically represented as shown in Fig.1.3.3.

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iii)  Consider the projection Q of the particle on the x-axis. Sketch a graph of the speed of Q against time t for one complete cycle.
 
The velocity of Q can be given by the slope of the x-t graph in Fig.1.3.2. 

Alternatively, differentiate 
wpe20.jpg (3258 bytes) 
The result is sketched as shown below:

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c0103c.gif (3745 bytes) 2
 
b. Ship A moves due east at a speed of 10ms-1. Ship B moves in direction N450E at a speed of 17ms-1. Sketch a diagram to show the velocity of A relative to B. Insert suitable numerical quantities in your diagram. 4 marks
 
c0103d.gif (4269 bytes) 2
The velocity of A relative to B is 
 wpe23.jpg (1193 bytes) 
as shown in Fig.1.3.5 

The magnitude is given by cosine rule: 
wpe24.jpg (4222 bytes)

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