Change in period due to a change in mass

In summary, we were tasked with finding an expression for ΔT, the small change in the period of a mass oscillating on a spring when the mass changes from m to m+Δm, where Δm << m. Using relevant equations and the binomial approximation, we arrived at the expression ΔT = T(1/2)(Δm/m), which makes sense as an increase in mass would result in an increase in period according to the original equations.
  • #1
Flipmeister
32
0

Homework Statement


A mass m oscillating on a spring has a period T. Suppose the mass changes very slightly from m to m+Δm, where Δm << m. Find an expression for ΔT, the small change in the period. Your expression should involve T, m, and Δm, but NOT the spring constant (k).


Homework Equations


[tex]T=\frac{2\pi }{\omega}=2\pi \sqrt{\frac{m}{k}}\\
\omega=\sqrt{\frac{k}{m}}[/tex]


The Attempt at a Solution


[tex]T+ΔT=2\pi \sqrt{\frac{m+Δm}{k}}[/tex]
Would that be the right way to start off? I can't figure out how to get rid of the k there. I don't see how conservation of energy can help, especially considering that conservation of energy is how those equations were derived in the first place...
 
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  • #2
You wrote a 'relevant equation' involving T, m and k, but you haven't used it.
 
  • #3
Flipmeister said:

The Attempt at a Solution


[tex]T+ΔT=2\pi \sqrt{\frac{m+Δm}{k}}[/tex]
Would that be the right way to start off? I can't figure out how to get rid of the k there. I don't see how conservation of energy can help, especially considering that conservation of energy is how those equations were derived in the first place...
Yes, this is a good start.
You need to use now the fact that Δm<<m and find an approximate expression of the RHS.
Do you know the expansion of [tex]\sqrt{1+x}[/tex] for small values of x?
 
  • #4
nasu said:
Yes, this is a good start.
You need to use now the fact that Δm<<m and find an approximate expression of the RHS.
Do you know the expansion of [tex]\sqrt{1+x}[/tex] for small values of x?

Yes, using the binomial approximation, I get ##\sqrt{m+Δm}=\sqrt{m} \sqrt{1+\frac{1}{2} \frac{Δm}{m}}##

haruspex said:
You wrote a 'relevant equation' involving T, m and k, but you haven't used it.
Hmm I suppose I should find T and ΔT separately next and plug those in...

[tex]T+ΔT=2π(1+\frac{Δm}{2m})(\sqrt{m/k})=2π\sqrt{m/k} + 2π\sqrt{Δm/k}
[/tex]

Simplifying I get √(m/k) = 2, so T=4π.

Solving for ΔT, I get ##ΔT=2π(1+(1/2) \frac{\Delta m}{m}) \sqrt{m/k} - T## which gets me...
$$\Delta T = 4π + 2π \frac{\Delta m}{m} -4π= T(\frac{1}{2} \frac{\Delta m}{m})$$

Does this look right? It seems to make sense (increasing the mass should increase period according to the starting equations). Is there any way for me to confirm this?
 
Last edited:
  • #5
Just realized this problem had a solution for it in the book, and that answer is right. Thanks for the help!
 
  • #6
Flipmeister said:
Simplifying I get √(m/k) = 2, so T=4π.

This step is not justified by the problem and you don't need this assumption to solve the problem. 2π√(m/k) = T, where T is the initial period.
However, the final result is OK.
 

1. How does the mass of an object affect its period?

The period of an object is directly proportional to the square root of its mass. This means that as the mass of an object increases, its period also increases.

2. Can the period of an object change without a change in mass?

Yes, the period of an object can also be affected by factors such as the length of its pendulum, the force of gravity, and the stiffness of the material it is attached to. These factors can cause the period to vary, even if the mass remains constant.

3. How does a change in mass affect the period of a pendulum?

The period of a pendulum is directly proportional to the square root of its length and inversely proportional to the square root of its mass. This means that as the mass of a pendulum increases, its period will increase, but if the length of the pendulum also increases, the period will decrease.

4. Does the type of material affect the period of an object?

Yes, the type of material can affect the period of an object. Objects made of stiffer materials will have a shorter period, while objects made of more flexible materials will have a longer period. This is because the stiffness of the material affects the frequency at which the object can oscillate.

5. How can we calculate the change in period due to a change in mass?

The change in period due to a change in mass can be calculated using the equation T=2π√(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity. By plugging in different values for mass and solving for T, we can determine how much the period will change for a given change in mass.

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