# M3 Simple Harmonic Motion

I'm having some trouble with shm questions. From my notes I've been told that a particle which moves on a straight line so that its acceleration is always towards a fixed point O in the line and is proportional to the displacement from O is said to move with SHM.

However, in many questions like the one below (to do with horizontal oscillations) the acceleration is away from the fixed point O. Can someone explain why this is and how you know which way the acceleration is going (i.e. towards or away from O). Also if you've got any tips on doing SHM questions then that'd be great.

So here's an example of the kind of question I've been getting stuck on (edexcel may 2004, qu 7):

A particle P of mass 0.3 kg is attached to one end of a light elastic spring. The other end of the spring is attached to a fixed point O on a smooth horizontal table. The spring has natural length 2 m and modulus of elasticity 21.6 N. The particle P is placed on the table at the point A, where OA = 2 m. The particle P is now pulled away from O to the point B, where OAB is a straight line with OB = 3.5 m. It is then released from rest.

(a) Prove that P moves with simple harmonic motion of period (pi/3)s

and the mark scheme says:

-(21.6x/2)=0.3a
-36x=a
SHM period=(2pi)/6=(pi)/3

so why is acceleration in the opposite direction to tension, and therefore away from O?
a=-w2 x

In simple harmonic motion acceleration is always proportional to minus displacement from a fixed origin.

If we resolve our direction to be negative to the left.

F=-36x=-w2 x

Equating gives

(w^2)=36

=>w=6

T=(2&#960;/W)

=>T=(2&#960;/6)

=>T=(&#960;/3)

Newton.
I'm having some trouble with shm questions. From my notes I've been told that a particle which moves on a straight line so that its acceleration is always towards a fixed point O in the line and is proportional to the displacement from O is said to move with SHM.

However, in many questions like the one below (to do with horizontal oscillations) the acceleration is away from the fixed point O. Can someone explain why this is and how you know which way the acceleration is going (i.e. towards or away from O). Also if you've got any tips on doing SHM questions then that'd be great.

The force is always towards a fixed point, the equilibrium position, but this point isn't necessarily the origin! You should always get an equation of the style a = -w^2*x where a is acceleration, w is frequency of oscillation, x is displacement from equilibrium position.
So here's an example of the kind of question I've been getting stuck on (edexcel may 2004, qu 7):

A particle P of mass 0.3 kg is attached to one end of a light elastic spring. The other end of the spring is attached to a fixed point O on a smooth horizontal table. The spring has natural length 2 m and modulus of elasticity 21.6 N. The particle P is placed on the table at the point A, where OA = 2 m. The particle P is now pulled away from O to the point B, where OAB is a straight line with OB = 3.5 m. It is then released from rest.

(a) Prove that P moves with simple harmonic motion of period (pi/3)s

and the mark scheme says:

-(21.6x/2)=0.3a
-36x=a
SHM period=(2pi)/6=(pi)/3

so why is acceleration in the opposite direction to tension, and therefore away from O?

The acceleration is in the same direction as the tension since it's the only force, but this force will be acting in the opposite direction from x so if you pull the particle more than 2m from O (ie in the +x direction), the force is in the -x direction, and vice versa
Thanks for the help
I'm having some trouble with shm questions. From my notes I've been told that a particle which moves on a straight line so that its acceleration is always towards a fixed point O in the line and is proportional to the displacement from O is said to move with SHM.

However, in many questions like the one below (to do with horizontal oscillations) the acceleration is away from the fixed point O. Can someone explain why this is and how you know which way the acceleration is going (i.e. towards or away from O). Also if you've got any tips on doing SHM questions then that'd be great.

So here's an example of the kind of question I've been getting stuck on (edexcel may 2004, qu 7):

A particle P of mass 0.3 kg is attached to one end of a light elastic spring. The other end of the spring is attached to a fixed point O on a smooth horizontal table. The spring has natural length 2 m and modulus of elasticity 21.6 N. The particle P is placed on the table at the point A, where OA = 2 m. The particle P is now pulled away from O to the point B, where OAB is a straight line with OB = 3.5 m. It is then released from rest.

(a) Prove that P moves with simple harmonic motion of period (pi/3)s

and the mark scheme says:

-(21.6x/2)=0.3a
-36x=a
SHM period=(2pi)/6=(pi)/3

so why is acceleration in the opposite direction to tension, and therefore away from O?

You could just forget the theory and think of the situation. When you have an elastic string or spring, the tension produced by the extension or compression will propel the particle back towards the equilibrium point. So the acceleration is always towards the equilibrium point, i.e. towards O.

Aitch