SHM spring block system

In summary, Homework Equations state that the time period for a block of mass m to oscillate on a smooth horizontal surface is T=2 pi sqrt{frac{m}{k}} where k is the equivalent of the mass of the spring connected to the block. When the block is displaced slightly, two springs connected to the block will push and pull it until it reaches the mean position. If the springs are less nice and make the restoring force unequal for each side, then the block will oscillate in a half-oscillatory motion.
  • #1
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Homework Statement



a block of mass m as shown in figure lies on a smooth horizontal surface and springs are at natural length. when the block is displaced slightly then find the time period of oscillation
20150206_235851-1.jpg

Homework Equations


$$T=2\pi\sqrt{\frac{m}{k}}$$
for series connection of springs:##\frac{k_1k_2}{k€1+k_2}##
for parellel connection:##k_1+k€2##

The Attempt at a Solution


for first 2 springs from top left,springs connected to block, equvivalent k is ##k##.
for bottom-most 2 springs 3k and 6k, k equivalent is ##2k## and you again have ##2k## for the bottom right corner spring.
how does the separation between block and spring affect the time period?
when I tried solving the problem by neglecting the detached springs, I got wrong answer. so what is the concept behind this? how should I proceed in such tricky cases?

answer:##2\pi\sqrt{\frac{m}{7k}}##
 
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  • #2
I imagine the "small separation" as meaning that it is touching the block, but it is not attached to the block.
In other words, it can push the block but it cannot pull the block
 
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  • #3
Nathanael said:
I imagine the "small separation" as meaning that it is touching the block, but it is not attached to the block.
In other words, it can push the block but it cannot pull the block
understood. but how will I find the time period? how should I use this informatiom?
 
  • #4
AdityaDev said:
understood. but how will I find the time period? how should I use this informatiom?
What is the restoring force when the block is displaced to the right? What is the restoring force when the block is displaced to the left?
 
  • #5
@Nathanael , I tried solving it. please verify.

you will be left with a spring of spring constant 6k and two detached springs of spring constant k. but all 3 springs are in parallel.
the 6k springs acts throughout.

if I displace the block by x distance to left and start measuring time, first both springs 6k and k push the block till the block reaches mean position and then k loses contact.
so ##t=T/4## from extreme to mean position.

from mean to right extreme, again ##t=T/4##

from right extreme to mean, T/4 and again T/4 to reach starting point.

here ##T=2\pi\sqrt{\frac{m}{7k}}## since from any mean to extreme, the effective spring constant is k+6k and since the separation is very small, I assumed that k springs act till mean position is reached.
 
  • #6
Nathanael said:
What is the restoring force when the block is displaced to the right? What is the restoring force when the block is displaced to the left?
7kx on both sides.
 
  • #7
Nathanael said:
What is the restoring force when the block is displaced to the right? What is the restoring force when the block is displaced to the left?
can you check if what I did was correct?
 
  • #8
AdityaDev said:
7kx on both sides.
Right. They made it easy by having it be equal on both sides. Since it's equal on both sides, you can effectively treat it as if it's attached to a single spring of constant 7k.

If they were less nice and made the restoring force unequal for each side, then I would proceed like you did in post #5.
(Although you don't need to look at quarter-oscillations, you can just look at half-oscillations. The total period would be half the time period for the motion due to one restoring force plus half the time period for the motion from the other restoring force.)

What you did is correct :)
 
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1. What is SHM (Simple Harmonic Motion)?

SHM, or Simple Harmonic Motion, refers to the back and forth movement of an object about an equilibrium point, caused by a restoring force that is proportional to the object's displacement from the equilibrium point.

2. What is a spring block system?

A spring block system is a physical system consisting of a mass attached to a spring, with the other end of the spring attached to a fixed point. The mass is free to move back and forth along a straight line, creating a simple harmonic motion.

3. How is the period of a spring block system calculated?

The period of a spring block system can be calculated using the equation T = 2π√(m/k), where T is the period, m is the mass of the object, and k is the spring constant.

4. What factors affect the frequency of a spring block system?

The frequency of a spring block system is affected by the mass of the object, the spring constant, and the amplitude (maximum displacement) of the oscillation.

5. How does the energy of a spring block system change during SHM?

The energy of a spring block system remains constant during SHM. As the object moves closer to the equilibrium point, the potential energy decreases and the kinetic energy increases, and vice versa as the object moves away from the equilibrium point.

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