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  • If displacement is sine function, then velocity function is cosine function and vice-versa.
  • The range of velocity lies between “-ωA” and “ωA”.
  • The velocity attains maximum value two times in a cycle at the center – (i) moving from negative to positive extreme and then (ii) moving from positive to negative extreme.
  • The velocity at extreme positions is zero.

Acceleration

The acceleration in linear motion is given as :

a = - ω 2 x = - k m x

Substituting for displacement “x”, we get an expression in variable time, “t” :

a = - ω 2 x = - ω 2 A sin ω t + φ

We can obtain this relation also by differentiating displacement function twice or by differentiating velocity function once with respect to time. Few important points about the nature of acceleration should be kept in mind :

1: Acceleration changes its direction about point of oscillation. It is always directed towards the center whatever be the position of the particle executing SHM.

2: Acceleration linearly varies with negative of displacement. We have seen that force-displacement plot is a straight line. Hence, acceleration – displacement plot is also a straight line. It is positive when “x” is negative and it is negative when “x” is positive.

Acceleration - displacement plot

The acceleration - displacement is a straight line bounded between two values.

3: Nature of force with respect to time, however, is not linear. If we combine the expression of acceleration and displacement, then we have :

a = - ω 2 x = - ω 2 A sin ω t + φ

Here, we draw both displacement and acceleration plots with respect to time in order to compare how acceleration varies as particle is at different positions.

Acceleration - time plot

The acceleration is represented by negative sine function.

The upper figure is displacement – time plot, whereas lower figure is acceleration – time plot. We observe following important points about variation of acceleration :

  • If displacement is sine function, then acceleration function is also sine function, but with a negative sign.
  • The range of acceleration lies between “ - ω 2 A ” and “ ω 2 A ”.
  • The acceleration attains maximum value at the extremes.
  • The acceleration at the center is zero.

4: Since force is equal to product of mass and acceleration, F = ma, it is imperative that nature of force is similar to that of acceleration. It is given by :

F = m a = - k x = - m ω 2 x = - m ω 2 A sin ω t + φ

Frequency, angular frequency and time period

The angular frequency is given by :

ω = k m = | a x | = | acceleration displacement |

We have used the fact " F = m a = - k x " to write different relations as above.

Time period is obtained from the defining relation :

T = 2 π ω = 2 π m k = 2 π | x a | = | displacement acceleration |

Frequency is obtained from the defining relation :

ν = 1 T = ω 2 π = 1 2 π k m = 1 2 π | a x | = | acceleration displacement |

Kinetic energy

The instantaneous kinetic energy of oscillating particle is obtained from the defining equation of kinetic energy as :

K = 1 2 m v 2 = 1 2 m ω 2 A 2 x 2 = 1 2 k A 2 x 2

The maximum value of KE corresponds to position when speed has maximum value. At x = 0,

K max = 1 2 m ω 2 A 2 0 2 = 1 2 k A 2

The minimum value of KE corresponds to position when speed has minimum value. At x = A,

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
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emma Reply
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what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
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Maurice Reply
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answer
Magreth
progressive wave
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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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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