Period of a child on a swing

In summary, when a child gets up from a sitting position to standing while swinging, the period changes due to a decrease in the distance between the pivot point and the center of mass. This can be treated as a simple pendulum, but in reality, a child on a swing cannot be approximated as such. The moment of inertia also changes, but assuming the worst case scenario, where the child's mass is distributed along a rod, the decrease in distance is still greater than the decrease in the moment of inertia, resulting in an overall increase in period.
  • #1
MuonMinus
8
0

Homework Statement


Question: If a child gets up from sitting position to standing while swinging, how does the period change?

Homework Equations


Period of a physical pendulum: T = 2π√(I/mgL), where I is the moment of inertia and L is the distance between the pivot and center of mass
Period of a simple (mathematical) pendulum: T = 2π√(L/g), where L is the distance between the (point) mass and the pivot

The Attempt at a Solution


The suggested answer I have seen is that a child on a swing is a physical pendulum. When the a child gets up, his center of mass moves up closer to the pivot point, so L (see the equation above) decreases, and the period therefore increases.
The problem I have with this answer is that when child gets up, his/her moment of inertia changes as well - how this can be taken into consideration?

Another possible answer is to consider the child a simple pendulum, in which case, when he gets up, L decreases and the period also decreases. But, in a real world, a child on a swing cannot be approximated by a simple pendulum!

How should this question be approached?
 
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  • #2
I would treat the child as a simple pendulum, an ideal case.

In the real world a child on a swing is something to enjoy because it gives you some time to relax.
 
  • #3
jedishrfu said:
I would treat the child as a simple pendulum, an ideal case.
This is probably how it was meant to be done. I just did not think it would be a reasonable approximation.

jedishrfu said:
In the real world a child on a swing is something to enjoy because it gives you some time to relax.

That's only if the child is old enough. Otherwise, it is a way to get exercise (by pushing), contemplating forced oscillations.
 
  • #4
The problem I have with this answer is that when child gets up, his/her moment of inertia changes as well - how this can be taken into consideration?
Key is the word "up". With ##I = \int r^2 dm## and ##L = \int r dm## it becomes clear that ## I/(mL) ## decreases when changing from sitting to standing.

Might need the parallel axis theorem to finish this off: worst case is changing from point mass at ##L## (swing length), so ##I = mL^2##, to a rod of length ##l## at ##L - l/2##: $$(L-l/2)^2 + l^2/12 < L^2 \ \ ?$$ leads to ## l(l-3L) < 0 ## which we can assume true.
 
  • #5

I would approach this question by considering the underlying principles and assumptions behind the equations given. In this case, the equations provided are for the period of a physical pendulum and a simple pendulum, which are idealized models that may not fully capture the dynamics of a child on a swing.

To address the issue of the changing moment of inertia, we would need to consider the distribution of mass and how it changes when the child gets up from a sitting to standing position. This would require more detailed measurements and calculations, taking into account the child's body shape and movements.

Furthermore, the equations provided assume that the swing is a perfect pendulum, with no external forces acting on it. In reality, there are air resistance, friction, and other factors that can affect the period of the swing.

In conclusion, while the equations provided can give us a rough estimate of how the period of a child on a swing may change when they get up, they do not fully capture the complexity of the system. More detailed measurements and calculations would be needed to accurately predict the change in period.
 

1. What is the period of a child on a swing?

The period of a child on a swing refers to the amount of time it takes for the swing to complete one full back-and-forth motion, also known as a cycle.

2. Does the length of the swing affect the period?

Yes, the length of the swing does affect the period. The longer the swing, the longer the period will be.

3. How does the weight of the child impact the swing period?

The weight of the child does not significantly impact the swing period. The period is primarily determined by the length of the swing and the gravitational force.

4. Can the period of a child on a swing be calculated?

Yes, the period of a child on a swing can be calculated using the formula T = 2π√(L/g), where T is the period, L is the length of the swing, and g is the gravitational acceleration (9.8 m/s^2).

5. Are there any factors that can affect the period of a child on a swing?

Aside from the length of the swing, other factors that can affect the period of a child on a swing include air resistance, the angle at which the child is swinging, and the strength and consistency of the child's pushes.

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