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Angular acceleration, α = - m g l I θ ------ for rotational SHM of pendulum

where “m” and “I” are mass and moment of inertia of the oscillating objects in two systems.

In order to understand the nature of cause, we focus on the block-spring system. When the block is to the right of origin, “x” is positive and restoring spring force is negative. This means that restoring force (resulting from elongation of the spring) is directed left towards the neutral position (center of oscillation). This force accelerates the block towards the center. As a result, the block picks up velocity till it reaches the center.

The plot, here, depicts nature of force about the center of oscillation bounded between maximum displacements on either side.

Nature of restoring force

Restoring force(s) tends to restore equilibrium.

As the block moves past the center, “x” is negative and force is positive. This means that restoring force (resulting from compression of the spring) is directed right towards the center. The acceleration is positive, but opposite to direction of velocity. As such restoring force decelerates the block.

Block-spring system

Restoring force(s) tends to restore equilibrium.

In the nutshell, after the block is released at one extreme, it moves, first, with acceleration up to the center and then moves beyond center towards left with deceleration till velocity becomes zero at the opposite extreme. It is clear that block has maximum velocity at the center and least at the extreme positions (zero).

From the discussion, the characterizing aspects of the restoring force responsible for SHM are :

  • The restoring force is always directed towards the center of oscillation.
  • The restoring force changes direction across the center.
  • The restoring force first accelerates the object till it reaches the center and then decelerates the object till it reaches the other extreme.
  • The process of “acceleration” and “deceleration” keeps alternating in each half of the motion.

Differential form of shm equation

We observed that acceleration of the object in SHM is proportional to negative of displacement. Here, we shall formulate the general equation for SHM in linear motion with the understanding that same can be extended to SHM along curved path. In that case, we only need to replace linear quantities with corresponding angular quantities.

a = - ω 2 x

where “ ω 2 ” is a constant. The constant “ω” turns out to be angular frequency of SHM. This equation is the basic equation for SHM. For block-spring system, it can be seen that :

ω = k m

where “k” is the spring constant and “m” is the mass of the oscillating block. We can, now, write acceleration as differential,

2 x t 2 = - ω 2 x

2 x t 2 + ω 2 x = 0

This is the SHM equation in differential form for linear oscillation. A corresponding equation of motion in the context of angular SHM is :

2 θ t 2 + ω 2 θ = 0

where "θ" is the angular displacement.

Solution of shm differential equation

In order to solve the differential equation, we consider position of the oscillating object at an initial displacement x 0 at t =0. We need to emphasize that “ x 0 ” is initial position – not the extreme position, which is equal to amplitude “A”. Let

t = 0, x = x 0, v = v 0

We shall solve this equation in two parts. We shall first solve equation of motion for the velocity as acceleration can be written as differentiation of velocity w.r.t to time. Once, we have the expression for velocity, we can solve velocity equation to obtain a relation for displacement as its derivative w.r.t time is equal to velocity.

Velocity

We write SHM equation as differential of velocity :

a = v t = = - ω 2 x

v x X x t = - ω 2 x

v v x = - ω 2 x

Arranging terms with same variable on either side, we have :

v v = - ω 2 x x

Integrating on either side between interval, while keeping constant out of the integral sign :

v 0 v v v = - ω 2 x 0 x x x

[ v 2 2 ] v 0 v = - ω 2 [ x 2 2 ] x 0 x

v 2 v 0 2 = - ω 2 x 2 x 0 2

v = { v 0 2 + ω 2 x 0 2 ω 2 x 2 }

v = ω { v 0 2 ω 2 + x 0 2 x 2 }

We put v 0 2 ω 2 + x 0 2 = A 2 . We shall see that “A” turns out to be the amplitude of SHM.

v = ω A 2 x 2

This is the equation of velocity. When x = A or -A,

v = ω A 2 A 2 = 0

when x = 0,

v max = ω A 2 0 2 = ω A

Displacement

We write velocity as differential of displacement :

v = x t = ω A 2 x 2

Arranging terms with same variable on either side, we have :

x A 2 x 2 = ω t

Integrating on either side between interval, while keeping constant out of the integral sign :

x 0 x x A 2 x 2 = ω 0 t t

[ sin 1 x A ] x 0 x = ω t

sin 1 x A sin 1 x 0 A = ω t

Let sin 1 x 0 A = φ . We shall see that “φ” turns out to be the phase constant of SHM.

sin 1 x A = ω t + φ

x = A sin ω t + φ

This is one of solutions of the differential equation. We can check this by differentiating this equation twice with respect to time to yield equation of motion :

x t = A ω cos ω t + φ

2 x t 2 = A ω 2 sin ω t + φ = ω 2 x

2 x t 2 + ω 2 x = 0

Similarly, it is found that equation of displacement in cosine form, x = A cos ω t + φ , also satisfies the equation of motion. As such, we can use either of two forms to represent displacement in SHM. Further, we can write general solution of the equation as :

x = A sin ω t + B cos ω t

This equation can be reduced to single sine or cosine function with appropriate substitution.

Example

Problem 1: Find the time taken by a particle executing SHM in going from mean position to half the amplitude. The time period of oscillation is 2 s.

Solution : Employing expression for displacement, we have :

x = A sin ω t

We have deliberately used sine function to represent displacement as we are required to determine time for displacement from mean position to a certain point. We could ofcourse stick with cosine function, but then we would need to add a phase constant “π/2” or “-π/2”. The two approach yields the same expression of displacement as above.

Now, according to question,

A 2 = A sin ω t

sin ω t = 1 2 = sin π 6

ω t = π 6

t = π 6 ω = π T 6 X 2 π = T 12 = 2 12 = 1 6 s

Questions & Answers

what is math number
Tric Reply
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
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12, 17, 22.... 25th term
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12, 17, 22.... 25th term
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I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
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Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
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can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
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I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
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Commplementary angles
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Nharnhar
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
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Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
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how do you translate this in Algebraic Expressions
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why surface tension is zero at critical temperature
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Source:  OpenStax, Oscillation and wave motion. OpenStax CNX. Apr 19, 2008 Download for free at http://cnx.org/content/col10493/1.12
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