Two masses connected by spring, find period of oscillation

In summary, the period of oscillation in this system can be found by considering the two masses connected by a spring and treating them as two separate oscillators with different spring constants. The period can be determined by using the concept of reduced mass and converting the two-body problem into a one-body problem.
  • #1
Aziza
190
1

Homework Statement



Two masses are connected by spring and slide freely without friction along horizontal track. What is period of oscillation?

Homework Equations


The Attempt at a Solution



My solution:
let x1 be position of mass 1 (m1) and x2 be position of mass 2 (m2) and L be length of spring in equilibrium.
Then, the total stretch of the spring is x2-x1-L. Also, F1 = -F2. Thus:

m1(x1)'' = -k(x1 - x2 + L)
m2(x2)'' = -k(x2 - x1 - L)

Solving for x2 from first eqn and substituting back into second eqn yield:

[itex]\frac{d^2}{dt^2}[\frac{m1 m2}{k}(x1)''+(m1+m2)x1][/itex] = 0

I am unsure how to proceed from here, any hints? I would like to just multiply both sides by (dt^2)/d^2 but I am unsure if this is mathematically correct? It does simplify the problem though and gives me right answer...
 
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  • #2
You should be well aware that ##\frac{d^2}{dt^2}## isn't a fraction.
 
  • #3
I don't think you really need differential equations for this one, since there is no external force there is a point on the spring that neither stretches nor compresses. on either side of this point is 2 springs with different spring constants, so effectively you have two different oscillators, with the same period.
 
  • #4
Do you know about reduced mass and how to convert a two-body problem into a one-body problem?
 
  • #5


To find the period of oscillation, you can use the formula T = 2π√(m/k), where T is the period, m is the total mass (m1 + m2), and k is the spring constant. In this case, since the masses are connected by the spring, the total mass is m1 + m2.

Another approach is to use the conservation of energy principle. At the equilibrium point, the total energy is all potential energy stored in the spring. As the masses oscillate back and forth, this potential energy is converted into kinetic energy and back to potential energy. The period of oscillation is the time it takes for the system to complete one full cycle of converting all potential energy into kinetic energy and back to potential energy.

To use this approach, you can set up the following equations:

At the equilibrium point:
Potential energy = 1/2k(x2-x1-L)^2

At the maximum displacement point:
Kinetic energy = 1/2m1(x1')^2 + 1/2m2(x2')^2 = 1/2m1(x1')^2 + 1/2m2(-x1')^2 = 1/2(x1')^2(m1+m2)

Equating these two energies and solving for x1' gives:
x1' = ±√(k/m1)(x2-x1-L)

Substituting this back into the equation for kinetic energy at the maximum displacement point gives:
Kinetic energy = 1/2(x1')^2(m1+m2) = 1/2(k/m1)(x2-x1-L)^2(m1+m2)

Setting this equal to the potential energy at the equilibrium point and solving for the period T gives:
T = 2π√(m1+m2)/k

Both methods should give you the same result for the period of oscillation.
 

FAQ: Two masses connected by spring, find period of oscillation

1. How do you calculate the period of oscillation for two masses connected by a spring?

The period of oscillation for two masses connected by a spring can be calculated using the equation T = 2π√(m/k), where T is the period, m is the mass of the system, and k is the spring constant. This equation assumes that there is no external force acting on the system and that the mass on the spring is negligible compared to the masses connected by the spring.

2. What is the significance of the period of oscillation in a system with two masses connected by a spring?

The period of oscillation is the time it takes for the system to complete one full cycle of oscillation. It is an important characteristic of the system as it determines the frequency and the amplitude of the oscillations. It also provides information about the stability and behavior of the system.

3. How does the mass and spring constant affect the period of oscillation in a system?

The period of oscillation is directly proportional to the square root of the mass and inversely proportional to the square root of the spring constant. This means that as the mass increases, the period of oscillation also increases, while a higher spring constant leads to a shorter period of oscillation.

4. What are the assumptions made when calculating the period of oscillation for two masses connected by a spring?

The calculation of the period of oscillation assumes that there are no external forces acting on the system, the spring is ideal and has no mass, and the masses connected to the spring do not affect its motion. Additionally, the calculation assumes that the system is in simple harmonic motion, meaning that the restoring force is directly proportional to the displacement from equilibrium.

5. Can the period of oscillation change in a system with two masses connected by a spring?

Yes, the period of oscillation can change in a system with two masses connected by a spring if there are changes in the mass, spring constant, or external forces acting on the system. For example, increasing the mass or decreasing the spring constant will result in a longer period of oscillation, while adding external forces can disrupt the simple harmonic motion and change the period.

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