Energy of an oscillating system

[AznBoi]

State how the total energy of an oscillating system depends on the amplitude of the motion, sketch or identify a graph of kinetic or potential energy as a function of time, and identify points in the motion where this energy is all potential or all kinetic.

I know that mechanical energy is conserved within an isloated system so $$\frac{1}{2} kx^{2}$$ can be used to represent the total energy of the system.

However, for the "sketch" part.. Does the energy vs. time graph look like a sine wave? The graph never reaches the negative y region right? Energy can never be negative right? Thanks.

 

[e(ho0n3]

As far as I know, energy is always a non-negative quantity. The potential and kinetic energy graphs will look sinusoidal.

 

[AznBoi]

Also, I'm not quire sure about this Quick Quiz quesiton, please help me out. Thanks!

Question: When an object moving in simple harmonic motion is at its maximum displacement from equilibrium, which of the following is at a maximum? a) velocity, b) acceleration, c) kinetic energy.

The answer is b) acceleration but I'm not sure why. Does it have to do with Newton's second law? Please explain.

 

[e(ho0n3]

Yes, it does have to do with Newton's second law. How does the force on the object undergoing simple harmonic motion depend on the displacement?

 

[AznBoi]

Yes, it does have to do with Newton's second law. How does the force on the object undergoing simple harmonic motion depend on the displacement?

The force: F=-kx of the spring is maximum when the displacement of the mass is maximum. So would the equation for acceleration be: $$a= \frac{kx-mg}{m}$$?? You need to include the "mg" also right? So since x is directly proportional to the acceleration then that means the maximum displacement of the mass means the most acceleration.

But at the greatest displacement, wouldn't the spring force equal the weight of the object? or am I getting confused? If the velocity is zero at that instant, wouldn't the acceleration be zero? I've had trouble with this concept before and I really would like to get it cleared up. Thanks!

 

[e(ho0n3]

Where did you get the "mg" term from? Perhaps you forgot to give us details about the problem in question. Also, if both the velocity and the acceleration were zero, the object would not move at all, i.e. no simple harmonic motion.

 

[AznBoi]

Where did you get the "mg" term from? Perhaps you forgot to give us details about the problem in question. Also, if both the velocity and the acceleration were zero, the object would not move at all, i.e. no simple harmonic motion.

When the spring is completely stretched at its amplitude, does its weight equal its spring force? I drew a free body diagram and made -kx equal to the weight of the mass (mg). $$F_{s} - W=ma$$ -->$$kx-mg=ma$$

I'm just confused on how there is an acceleration at the maximum displacement from equilibrium. There is no motion right? But the spring force is greater than the weight?

 

[e(ho0n3]

When the spring is completely stretched at its amplitude, does its weight equal its spring force?

I guess you're talking about a mass hanging from a spring, the whole of which is vertical to the ground. When the mass is displaced a maximum distance from its equilibrium position, the force exerted by the spring on the mass is at its maximum. If it were equal to the weight of the mass, the mass would not oscillate at all. It would just stay still.

I drew a free body diagram and made -kx equal to the weight of the mass (mg). $$F_{s} - W=ma$$ -->$$kx-mg=ma$$

I'm just confused on how there is an acceleration at the maximum displacement from equilibrium. There is no motion right? But the spring force is greater than the weight?

There is an instance in time in which the mass is not moving, i.e. when the mass is at maximum displacement from the equilibrium position. However, the acceleration at this instance is not 0 and so the mass will start moving again thenafter.

 

[bdrosd]

There need not be any velocity for there to be a non-zero acceleration. The most common example of this is when a ball is thrown vertically upwards and momentarily had zero velocity at the top of its travel. The moment the ball left the thrower's hand, it was accelerating downward, due to gravity. This statement is true while the ball was going up; at the peak; and while coming back down. Acceleration is the rate of change of velocity, so if an object in SHM is reversing direction at the maximum extent of its travel, as it goes from (+) to (-) velocity, it is definitely accelerating, yet has zero velocity. In fact the acceleration at this particular point is larger than any other time during SHM, which is consistent with the fact that the restoring force is largest when the object is a maximum displacement from the equilibrium position.

 

[AznBoi]

Oh.. I see now. Thanks to both of you! =]