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Mechanics, EXT2 M1 2020 HSC 13a

A particle is undergoing simple harmonic motion with period . The central point of motion of the particle is at  . When    the particle has its maximum displacement of    from the central point of motion.

Find an equation for the displacement, , of the particle in terms of .    (3 marks)

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Mechanics, EXT2* M1 2018 HSC 10 MC

A particle is moving in simple harmonic motion. The displacement of the particle is    and its velocity, , is given by the equation  , where and are constants.

The particle is initially at  .

Which function, in terms of time , could represent the motion of the particle?

A.     

B.     

C.     

D.     

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♦♦ Mean mark 33%.

   

   
 

Mechanics, EXT2* M1 2016 HSC 13a

The tide can be modelled using simple harmonic motion.

At a particular location, the high tide is 9 metres and the low tide is 1 metre.

At this location the tide completes 2 full periods every 25 hours.

Let be the time in hours after the first high tide today.

  1. Explain why the tide can be modelled by the function  .  (2 marks)

  2. The first high tide tomorrow is at 2 am.

     

    What is the earliest time tomorrow at which the tide is increasing at the fastest rate?  (2 marks)

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i.   
 

 

 

 

 

 

ii.   

 

 

 
 

 

Mechanics, EXT2* M1 2004 HSC 7a

The rise and fall of the tide is assumed to be simple harmonic, with the time between successive high tides being 12.5 hours. A ship is to sail from a wharf to the harbour entrance and then out to sea. On the morning the ship is to sail, high tide at the wharf occurs at 2 am. The water depths at the wharf at high tide and low tide are 10 metres and 4 metres respectively.

  1. Show that the water depth, metres, at the wharf is given by
     
         , where    is the number of hours after high tide.  (2 marks)  

  2. An overhead power cable obstructs the ship’s exit from the wharf. The ship can only leave if the water depth at the wharf is 8.5 metres or less.

     

    Show that the earliest possible time that the ship can leave the wharf is 4:05 am.  (2 marks)

  3. At the harbour entrance, the difference between the water level at high tide and low tide is also 6 metres. However, tides at the harbour entrance occur 1 hour earlier than at the wharf. In order for the ship to be able to sail through the shallow harbour entrance, the water level must be at least 2 metres above the low tide level.

     

    The ship takes 20 minutes to sail from the wharf to the harbour entrance and it must be out to sea by 7 am. What is the latest time the ship can leave the wharf?  (2 marks)

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i.   

 

 

 

 

 

  

ii.  

 Calculus in the Physical, EXT1 2004 HSC 7a Answer

 
 

 

 

iii. 

 
 

 

 

Mechanics, EXT2* M1 2007 HSC 6a

A particle moves in a straight line. Its displacement, metres, after seconds is given by

.

  1. Prove that the particle is moving in simple harmonic motion about    by showing that   .  (2 marks)

  2. What is the period of the motion?  (1 mark)

  3. Express the velocity of the particle in the form  α, where  α  is in radians.  (2 marks)

  4. Hence, or otherwise, find all times within the first    seconds when the particle is moving at metres per second in either direction.  (2 marks)

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i.   

 
 
 

 

ii. 

 

 

iii. 

α

αα
αα
α
α
 

 

α
α

 

(iv) 

 

 

Mechanics, EXT2* M1 2015 HSC 13a

A particle is moving along the -axis in simple harmonic motion. The displacement of the particle is metres and its velocity is ms. The parabola below shows as a function of .

2015 13a

  1. For what value(s) of is the particle at rest?  (1 mark)

  2. What is the maximum speed of the particle?  (1 mark)

  3. The velocity of the particle is given by the equation

           where , and are positive constants.

     

    What are the values of , and ?  (3 marks)

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i.   

 

ii.   

 

iii. 

♦ Mean mark 41%.

 
 

Mechanics, EXT2* M1 2009 HSC 5a

The equation of motion for a particle moving in simple harmonic motion is given by

where    is a positive constant,    is the displacement of the particle and    is time.  

  1. Show that the square of the velocity of the particle is given by
     
         

     

    where    and    is the amplitude of the motion.   (3 marks)

  2. Find the maximum speed of the particle.     (1 mark)

  3. Find the maximum acceleration of the particle.    (1 mark)

  4. The particle is initially at the origin. Write down a formula for    as a function of  , and hence find the first time that the particle’s speed is half its maximum speed.   (2 marks)

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i.   

 

 
 

 

ii.   
 

 

iii.  
 

 

iv.  
 

 

Mechanics, EXT2* M1 2013 HSC 12e

A particle moves along a straight line. The displacement of the particle from the origin is , and its velocity is . The particle is moving so that  , where is a constant.

Show that the particle moves in simple harmonic motion with period  .   (2 marks)

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Mechanics, EXT2* M1 2011 HSC 3a

The equation of motion for a particle undergoing simple harmonic motion is 

 ,

where is the displacement of the particle from the origin at time , and is a positive constant.

  1. Verify that  , where and are constants, is a solution of the equation of motion.    (1 mark)

  2. The particle is initially at the origin and moving with velocity . 

     

    Find the values of and in the solution  .    (2 marks)

  3. When is the particle first at its greatest distance from the origin?   (1 mark)

  4. What is the total distance the particle travels between    and  ?   (1 mark)

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i.  
 
 
   
   

 

ii.   

  
♦♦ Mean mark part (iii) 47%
 

iii.   

 

 

iv. 

♦♦ Mean mark 22%
MARKER’S COMMENT: Many students found the displacement at rather than the distance travelled while another common error found distance as 2 x amplitude.

 

 

Mechanics, EXT2* M1 2012 HSC 13c

A particle is moving in a straight line according to the equation

, 

where is the displacement in metres and is the time in seconds.

  1. Prove that the particle is moving in simple harmonic motion by showing that  satisfies an equation of the form  .  (2 marks)

  2. When is the displacement of the particle zero for the first time?    (3 marks)

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i. 

 
 
 

 

ii. 

♦ Mean mark 42%.
IMPORTANT: The critical insight required to solve is to realise that the cosine of the difference between 2 angles, i.e. , applies.

 

 Calculus in the Physical World, EXT1 2012 HSC 13c Answer