# TBR - Dampened Harmonic Oscillation

#### elloL

##### Full Member
Hello,

I am stuck on a question about the energy of a system that is undergoing dampened harmonic motion and I feel like the given answer may be wrong, although I could just be misunderstanding what it is asking. Basically, it asks about what a true observation about the system is. I selected that the potential energy at consecutive maxima decreases while the kinetic energy stays constant. At the amplitude of the object, I know that it has zero velocity because it is changing direction and if it is dampening, it is losing energy to its environment; therefore, wouldn't the potential energy decrease while the kinetic energy remains constant (or zero)? (This is on page 146 on the TBR physics book).

#### iamgroot2

##### Full Member
So, picture a ball (or some other weight) on a rope attached to the ceiling hook forming a pendulum. In a system, where there are no nonconservative forces (like friction) to dissipate the energy of our system, this is would result in a simple harmonic motion, meaning we go to a highest point away from its equilibrium position where we have our highest potential energy (mgh) but lowest kinetic energy (1/2*m*v^2), where at the highest point, kinetic energy is zero, due to velocity being zero. As our ball comes back down and reaches its original equilibrium position (at the bottom of its swing), it will have its highest kinetic energy and lowest potential energy (height is the smallest and velocity is the largest at this point, meaning potential energy is zero when kinetic energy is maximum), where it will then go to the other highest point away from its equilibrium position on the other side of its swing, where it will again go to having a max potential energy and min kinetic energy. This decrease in kinetic energy and increase in potential energy happens because the ball experiences a restoring force that points toward the equilibrium position, as it goes to either of its highest points from the equilibrium position (in terms of height), meaning it experiences deceleration or a decrease in kinetic energy (velocity decreases), while it gains height, so potential energy is gained. This means that energy in this simple harmonic motion with only conservative forces present is being converted between kinetic and potential energy. If we were to plot the time vs amplitude, we would see a sinusoidal wave pattern. This shows that the ball is following a path of going between two maxima in terms of potential energy over and over again which we call oscillation (meaning this can apply to any oscillating system, like a spring, that displays this behavior, as well).

When the ball is started off in a position going going in one direction, following a path of simple harmonic motion, and it returns to this position going in the same direction (i.e the equilibrium position going to the right), the wave on the graph we refer to as one wavelength in terms of distance between these two points. This wave segment (if you were the look just at the wave at and between these two points) is also called one complete cycle (it cycles back to its original position and motion). Period refers to the time the ball (or spring or some other object that follows this motion) takes to complete one circle (hence its units are in seconds). Frequency is the inverse of this and refers to the amount of cycles per second (hence its units are in seconds^-1 , but are also in Hertz). There is one more important aspect of a graphical wave, which usually pertains to the energy of the system, which we call amplitude, which is the displacement of the object from its equilibrium position.

So, in our simple harmonic motion without any nonconservative forces present, potential energy maxima will stay the same and the kinetic energy maximum will stay the same. We could theoretically see this with a pendulum going back and forth between two its highest points FOREVER.

However, in most physics classes when they do a demo using this pendulum ball system, this is not what happens. Instead, we see a ball released from some height, which then travels to its opposite side to its other highest potential energy maximum point and then back. But, slowly over time, the arc length continuously decreases and the ball swings back and forth by a smaller and smaller distance. Meaning, the amplitudes (displacements from equilibrium) become less and less over time displaying an attenuating pattern (the wave becomes more smushed or flattened towards the x-axis with each cycle). We call this pattern on the graph (or motion) dampened harmonic oscillation.

What can cause this?

The nonconservative forces of wind resistance hitting the ball and rope and the frictional forces between the ceiling rope and the hook, resulting in the energy within the pendulum system being given off, or lost, as heat. Remember, that friction in most cases, as in this pendulum case, is opposite or opposes the direction of motion. Meaning, the frictional forces are opposite the direction of pendulum acceleration, causing it to accelerate less than it would have had those frictional forces not been present from either of its highest points going towards its equilibrium position.

This means starting from one potential energy maxima, where height is the highest and going towards equilibrium, the potential energy will be converted to heat and kinetic energy at each point along its downward swing until it reaches equilibrium, with velocity will less than it would be had frictional forces not been present. This results in less kinetic energies at each point on the downward swing, compared to simple harmonic motion, resulting in a highest kinetic energy position at the equilibrium position that is less than what it was in the simple harmonic motion, due to some of the potential energy being converted to heat.

This lesser kinetic energy (in our energy savings bank account) would then be converted to lesser potential energy (funds available) gained reaching the other sides potential energy maxima (a smaller change in height), with energy being lost to heat along the way (more energy lost in the system to the opposing frictional forces). This results in a lesser potential energy on the other side, which than gets converted into lesser kinetic energy at the equilibrium, with energy again being lost to heat. This cycle repeats with the arc length getting smaller and smaller with both kinetic and potential energy maxima continuously decreasing, until the ball rests at the equilibrium position at 0 joules for kinetic and potential energy (all energy lost to heat). If you look at a graph, this would explain why the amplitudes continuously decrease with each cycle.

So, we know that kinetic energy is decreasing, with the average velocity of the pendulum continuously going down (with mass staying the same). However, because the arc distance is decreasing with each cycle, while velocity is also decreasing each cycle, the time it takes to complete each cycle will actually stay the same (velocity = displacement/time). This means that the period will stay the same (constant) and frequency being the inverse of the period and also being the cycles per second will also stay constant.

A final note is that if frequency is equal to the inverse of the period (or vice versa), both either change or both remain the same.

Last edited:
• 1 user

#### elloL

##### Full Member
So, picture a ball (or some other weight) on a rope attached to the ceiling hook forming a pendulum. In a system, where there are no nonconservative forces (like friction) to dissipate the energy of our system, this is would result in a simple harmonic motion, meaning we go to a highest point away from its equilibrium position where we have our highest potential energy (mgh) but lowest kinetic energy (1/2*m*v^2), where at the highest point, kinetic energy is zero, due to velocity being zero. As our ball comes back down and reaches its original equilibrium position (at the bottom of its swing), it will have its highest kinetic energy and lowest potential energy (height is the smallest and velocity is the largest at this point, meaning potential energy is zero when kinetic energy is maximum), where it will then go to the other highest point away from its equilibrium position on the other side of its swing, where it will again go to having a max potential energy and min kinetic energy. This decrease in kinetic energy and increase in potential energy happens because the ball experiences a restoring force that points toward the equilibrium position, as it goes to either of its highest points from the equilibrium position (in terms of height), meaning it experiences deceleration or a decrease in kinetic energy (velocity decreases), while it gains height, so potential energy is gained. This means that energy in this simple harmonic motion with only conservative forces present is being converted between kinetic and potential energy. If we were to plot the time vs amplitude, we would see a sinusoidal wave pattern. This shows that the ball is following a path of going between two maxima in terms of potential energy over and over again which we call oscillation (meaning this can apply to any oscillating system, like a spring, that displays this behavior, as well).

When the ball is started off in a position going going in one direction, following a path of simple harmonic motion, and it returns to this position going in the same direction (i.e the equilibrium position going to the right), the wave on the graph we refer to as one wavelength in terms of distance between these two points. This wave segment (if you were the look just at the wave at and between these two points) is also called one complete cycle (it cycles back to its original position and motion). Period refers to the time the ball (or spring or some other object that follows this motion) takes to complete one circle (hence its units are in seconds). Frequency is the inverse of this and refers to the amount of cycles per second (hence its units are in seconds^-1 , but are also in Hertz). There is one more important aspect of a graphical wave, which usually pertains to the energy of the system, which we call amplitude, which is the displacement of the object from its equilibrium position.

So, in our simple harmonic motion without any nonconservative forces present, potential energy maxima will stay the same and the kinetic energy maximum will stay the same. We could theoretically see this with a pendulum going back and forth between two its highest points FOREVER.

However, in most physics classes when they do a demo using this pendulum ball system, this is not what happens. Instead, we see a ball released from some height, which then travels to its opposite side to its other highest potential energy maximum point and then back. But, slowly over time, the arc length continuously decreases and the ball swings back and forth by a smaller and smaller distance. Meaning, the amplitudes (displacements from equilibrium) become less and less over time displaying an attenuating pattern (the wave becomes more smushed or flattened towards the x-axis with each cycle). We call this pattern on the graph (or motion) dampened harmonic oscillation.

What can cause this?

The nonconservative forces of wind resistance hitting the ball and rope and the frictional forces between the ceiling rope and the hook, resulting in the energy within the pendulum system being given off, or lost, as heat. Remember, that friction in most cases, as in this pendulum case, is opposite or opposes the direction of motion. Meaning, the frictional forces are opposite the direction of pendulum acceleration, causing it to accelerate less than it would have had those frictional forces not been present from either of its highest points going towards its equilibrium position.

This means starting from one potential energy maxima, where height is the highest and going towards equilibrium, the potential energy will be converted to heat and kinetic energy at each point along its downward swing until it reaches equilibrium, with velocity will less than it would be had frictional forces not been present. This results in less kinetic energies at each point on the downward swing, compared to simple harmonic motion, resulting in a highest kinetic energy position at the equilibrium position that is less than what it was in the simple harmonic motion, due to some of the potential energy being converted to heat.

This lesser kinetic energy (in our energy savings bank account) would then be converted to lesser potential energy (funds available) gained reaching the other sides potential energy maxima (a smaller change in height), with energy being lost to heat along the way (more energy lost in the system to the opposing frictional forces). This results in a lesser potential energy on the other side, which than gets converted into lesser kinetic energy at the equilibrium, with energy again being lost to heat. This cycle repeats with the arc length getting smaller and smaller with both kinetic and potential energy maxima continuously decreasing, until the ball rests at the equilibrium position at 0 joules for kinetic and potential energy (all energy lost to heat). If you look at a graph, this would explain why the amplitudes continuously decrease with each cycle.

So, we know that kinetic energy is decreasing, with the average velocity of the pendulum continuously going down (with mass staying the same). However, because the arc distance is decreasing with each cycle, while velocity is also decreasing each cycle, the time it takes to complete each cycle will actually stay the same (velocity = displacement/time). This means that the period will stay the same (constant) and frequency being the inverse of the period and also being the cycles per second will also stay constant.

A final note is that if frequency is equal to the inverse of the period (or vice versa), both either change or both remain the same.
Thanks for the response - I'm pausing my physics prep to focus on other topics, but I'll definitely read this over when I start again.

This thread is more than 1 year old.

Your message may be considered spam for the following reasons: