Harmonic Oscillator: Solving Newton's Second Law

In summary: If you are confused about the direction of the acceleration vector, think about how the speed of the object is changing. If the object is going faster, the acceleration vector will be directed more towards the center of the circle. If the object is slowing down, the acceleration vector will be directed more towards the periphery of the circle.
  • #1
induing
4
0
hello,

new here and confused about Newton second Law.

given:
vertical mass damper system, position of the mass: x(t)=sin(t)
velocity is: v(t)=cos(t)
acceleration is: a(t)=-sin(t)

function x(t): above x-axis describes position of the mass below the vertical equilibrium point, which (below) is the positive direction of vector x


suppose I look at the movement between t=0 and t=T/4: when the mass is below the vertical equilibrium line and is moving to the ground

when I apply Newton second law (ma(t)) the vector a(t) must point downward because mass is going down, but when I look at the the function a(t) is it negative, which means the vector a(t) is pointing upward (ponting in negative direction)...

I'm off the right track, but don't see my fault..

so if anyone can help, please?

grtz
 
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  • #2
If I understand this correctly, the acceleration vector should point upwards because, on the way down, the mass is approaching its lowest position and will have zero velocity there. This upward acceleration vector provides the necessary deceleration of the mass. The velocity vector points downward, however.
 
  • #3
so the term (ma(t)) isn't the vector which is related to the movement of the mass?

i find that rather weird, because ma(t) equals the vector sum of all the (external) forces, so i would pressume that this net force gives the direction...

and if I do understand this correctly, I cannot predict the movement of the mass just by looking at the acceleration vector?

thank you for reply!
grtz
 
  • #4
The movement of a body is generally represented by the velocity vector, which is always tangential to the motion. For example, if a car was traveling eastwards, the velocity vector would point eastwards.

The acceleration vector, however, acts in the direction opposite to motion if the object is slowing down. In the above example, if the car was traveling eastwards and slowed down to turn a corner,say, the velocity vector points eastwards, but the acceleration vector points westwards. We say the car has a negative acceleration (in my first post deceleration should be replaced with negative acceleration to be more precise). However, in accelerating a car, for example, the velocity and acceleration vectors do point in the same direction. (net force on car is in this direction). [tex] \sum F_x = ma_x [/tex]
 
  • #5
thanks for the clear and brief explanation!
must have been very confused: force which is needed to give a body certain amount of movement, is indeed the thing which control way the way the body change it's movement

more friction will slow it down down, so decceleration kicks in, but body off course is still moving some time in the same direction.

thank you big time,sir!
grtz
 
  • #6
Perhaps to give another example, have you ever looked at circular motion? The velocity vector is always tangent to the circle if a body is undergoing circular motion. But the acceleration vector can be in quite unexpected places. If there is no tangential acceleration (that is, constant velocity and uniform circular motion) then the acceleration vector is radially inwards (the body experiences a centripetal acceleration). However, if there is a tangential acceleration, the resultant acceleration vector is the vector sum of the centripetal acc. and tangential acc. and we get a vector that is slightly offset from being directed exactly towards the centre of the circle.
 
  • #7
yeah I know the dynamics of circular motion, constant velocity or accelerated
the normal (radial) component is needed to change movement of direction and is always presented

if you want a accelerated circular movement (speeding up in a curved road) the tangential vector which produces change in velocity kicks in

acceleration vector then consists of 2 vector.

have been confused with force direction as being the direction of the acceration vector, which just indicates the way the speed is changing an not the movement itself

soon examination of dynamics,
so many thanks again for the quick reply!

grtz
 

1. What is a Harmonic Oscillator?

A Harmonic Oscillator is a physical system that experiences a restoring force that is proportional to the displacement from its equilibrium position. This type of motion can be described by a sinusoidal function, hence the term "harmonic". Examples of harmonic oscillators include a mass on a spring or a pendulum.

2. How does Newton's Second Law relate to Harmonic Oscillators?

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In the case of a harmonic oscillator, the restoring force acts as the net force and the mass is the object experiencing the acceleration. This relationship allows us to mathematically model the motion of a harmonic oscillator using Newton's Second Law.

3. What is the equation for a Harmonic Oscillator?

The equation for a harmonic oscillator is given by: F = -kx, where F is the restoring force, k is the spring constant, and x is the displacement from the equilibrium position. This equation is derived from Newton's Second Law and is known as Hooke's Law.

4. How do you solve for the motion of a Harmonic Oscillator?

To solve for the motion of a harmonic oscillator, we can use the equation: x(t) = A*cos(ωt + ϕ), where A is the amplitude, ω is the angular frequency, and ϕ is the phase angle. These values can be determined using initial conditions, such as the initial displacement and velocity of the oscillator. This equation allows us to predict the position of the oscillator at any given time.

5. What are the applications of Harmonic Oscillators in science and engineering?

Harmonic oscillators have many practical applications in science and engineering. They are used to study and understand the behavior of systems in various fields such as physics, chemistry, and biology. In engineering, harmonic oscillators are used in the design of springs, shock absorbers, and vibration isolation systems. They are also used in technologies such as clocks, watches, and musical instruments.

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