A glider is oscillating in SHM on an air track with an amplitude A1 . You slow it so that its am

A glider is oscillating in SHM on an air track with an...

Key Concepts

Simple Harmonic Motion (SHM)

SHM is a type of periodic motion where the restoring force is directly proportional to displacement from an equilibrium position and acts in the opposite direction. This concept is central to understanding many oscillatory systems because it allows derivation of important relationships between displacement, velocity, acceleration, and energy.

Period and Frequency Independence from Amplitude

In SHM, the period (and thus frequency and angular frequency) is determined solely by the system’s parameters, such as mass and spring constant, and remains unchanged irrespective of the amplitude of oscillation. This invariance is a unique characteristic of ideal linear oscillators, meaning that reducing amplitude does not affect the timing of oscillations.

Mechanical Energy in SHM

The total mechanical energy in SHM, which is the sum of kinetic and potential energies, is proportional to the square of the amplitude. Consequently, if the amplitude is halved, the mechanical energy decreases by a factor of four. This quadratic dependence underlines the sensitivity of stored energy in oscillatory systems to changes in amplitude.

Maximum Speed in SHM

The maximum speed in an oscillatory motion occurs as the object passes through the equilibrium position and is given by the product of the angular frequency and the amplitude. A reduction in amplitude leads directly to a proportional decrease in maximum speed, illustrating the linear relationship between amplitude and peak velocity in SHM.

Energy Distribution at Intermediate Positions in SHM

At positions other than the extremes or equilibrium, the energy in SHM is divided between kinetic and potential forms. The potential energy is related to the square of the displacement, while the kinetic energy is what remains from the total mechanical energy. Understanding this energy partitioning helps predict how speeds and energies change as the oscillator moves, such as at a specific fraction of the amplitude.

Transcript

00:01 In the a part, we have to find out the dependency of amplitude with various quantity is given.

00:07 So the first is the time period and we know the formula for time period is 2 pi into root of m upon k.

00:16 Here you cannot find this is a constant.

00:19 There is no variable which represents amplitude.

00:23 Hence, this is independent of a.

00:25 Therefore, the value does not change.

00:29 The second one is frequency.

00:35 That is one upon t.

00:37 One is a constant.

00:38 It is only dependent on time period and we have already told the time period doesn't change with respect to amplitude.

00:45 Hence frequency also doesn't change.

00:51 The third is our angle of frequency and the formula we know that it is 2 pi f.

00:58 2 is a constant.

00:59 P is a constant that is only related to frequency and we have already told that it is not dependent on amplitude, hence angular frequency is also independent of amplitude, hence doesn't change.

01:16 So we have to understand the time period, frequency and angular frequency, all the three quantities are not dependent on the amplitude.

01:25 In the b part, we have been given energy and we know the formula that is e equal to half ka square.

01:36 When x is equal to plus or minus a now when a is half that is a becomes a by two what will be the new e one half k a by two the whole square that is half k a square upon four which is equal to e by four so what do we understand by this we understand that when a becomes a by 2 then he decreases by a factor of 4 so this is what happens when amplitude decreases by half energy decreases by a factor of 4 understood this much so let's move on to the c part the c part it's given b max that is we know the formula that is omega into a.

02:45 Omega we know 2 pi f into a.

02:49 Okay.

02:50 So, v max 1, that is equals to 2 pi f, a1.

02:59 And v max 2 equal to 2 pi f, a 2 .5a2.

03:05 Now, f doesn't change with respect to, f doesn't change with respect to amplitude.

03:12 Now since a2 equal to half a1, okay, since a2 is half a1, v max equal to v max 2 equal to 2 pi f, half a1, that is equal to half v max 1.

03:40 Because a2 is half of a1, v max will also be half of the maximum of the velocity of one.

03:49 Understood this much? so let's move on to the d part.

04:00 D part we have vx.

04:02 That is velocity is given by the formula plus or minus root of k upon m into root of a squared minus x square.

04:15 Now, x equal to, it is given plus or minus a1 upon 4...

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