# SHM child on a swing question

• blackcat
In summary, the max. kinetic energy of the oscillator is equal to the "spring constant for the pendulum" multiplied by the "oscillator's " "kinetic energy."

#### blackcat

Hi,

A child on a swing swings with a time period of 2.5s and an amplitude of 2m.

What is the max. kinetic energy of the oscillation?

I'm not sure how to work this out without her mass. Her max speed is 2.51m/s but I don't know how to do this. BTW this is all the information that is given

Any hints?

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Hint: Draw a picture and use the formula for period of a pendulum. Once you find the height between the highest and lowest points, you can find the kinetic energy.

During the SHM motion the oscillator continually converts potential energy to kinetic energy and back. At the extremes of its motion it momentarily comes to rest. At these points all energy is converted to potential energy. When the oscillator is at its equilibrium position all of its energy is converted back to kinetic energy. So try and find the maximum potential eneregy of the oscillator. It might be helpfull to totally forget that you are dealing with a swing and just concentrate on the maths. Anyway, the statement that the amplitude of the swing is 2 meters can be interpreted in many ways.

Ok thanks both of you.

What I was trying to say is that the maximum kinetic energy of the oscillator should be equal to

$$\frac{1}{2}kA^2$$

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html" [Broken]

so all you need is the "spring constant for the pendulum" - which unfortunately does depend on the mass!

http://theory.uwinnipeg.ca/physics/shm/node5.html" [Broken]

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Looking at it differently one can say

$$\Gamma = I \alpha$$

which gives

$$\ddot{\theta} = \frac{1}{I} \Gamma$$

for

$$\Gamma = lw \sin(\theta)$$

for small swing angles (which the condition for SHM for a pendulum) one gets

$$\Gamma = lmg\theta$$

which gives the more prommising (maybe?) SHM equation

$$\ddot{\theta} = -\frac{g}{l} \theta$$

the justification for inserting the - is that the torque is positive (anticlockwise) when the angle is negative (to the left of the equilibrium) and vice versa when the pendulum is on the other side of the equilibrium.

But I do'nt think that one can get pass the fact that the total energy of a pendulum do depend on the mass. For the spring not so. This can be understood on the basis that the energy is stored in totallity in the spring when it is strecthed (compressed) to its max, but for the pendulum the max energy depends on the mass swinging from it. If a larger mass swings up to the same height on the same length of string the total energy of the system will just be more. And as we all know the period does not depend on the mass, just the length.

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## 1. What is SHM (Simple Harmonic Motion) and how does it relate to a child on a swing?

SHM is a type of periodic motion in which an object moves back and forth around a central point. In the case of a child on a swing, the central point is the pivot point of the swing and the motion is generated by the child's weight and pushing force.

## 2. What factors affect the frequency of a child's swinging motion?

The length of the swing, the child's weight, and the force of the push all affect the frequency of the swinging motion. A longer swing, heavier child, and stronger push will result in a higher frequency (faster) swinging motion.

## 3. How does the amplitude of the swing change as the child swings?

The amplitude, or the maximum distance the child swings from the central point, decreases as the child swings due to the dissipation of energy through air resistance and friction at the pivot point.

## 4. What is the relationship between the period and frequency of a child's swinging motion?

The period, or the time it takes for the child to complete one swing, is inversely proportional to the frequency. This means that as the frequency increases, the period decreases and vice versa.

## 5. Can SHM be applied to other swinging objects, such as a pendulum?

Yes, SHM principles can be applied to any object that exhibits periodic motion around a central point. This includes pendulums, which have a similar motion to a child on a swing, but with a different pivot point and force causing the motion.