Question

The displacement of a S.H.O. is given by the equation \(\mathrm{x}=\mathrm{A}\) \(\cos \\{\omega t+(\pi / 8)\\}\). At what time will it attain maximum velocity? (A) \((3 \pi / 8 \omega)\) (B) \((8 \pi / 3 \omega)\) (C) \((3 \pi / 16 \omega)\) (D) \((\pi / 16 \pi)\).

Short answer

The time at which the S.H.O. will attain its maximum velocity is (C) \(\frac{3 \pi}{16 \omega}\).

Step-by-step solution

Identify the given equation

We have the displacement equation for a S.H.O.: x = A cos(ωt + π/8) where A is the amplitude, ω is the angular frequency, and t is the time.

Differentiate the displacement equation

In order to find the velocity equation, we need to differentiate the displacement equation with respect to time: \(v = \frac{dx}{dt} = \frac{d(A\cos(\omega t + \frac{\pi}{8}))}{dt}\) Now, apply the chain rule for the differentiation: \(v = -A\omega \sin(\omega t + \frac{\pi}{8})\)

Determine the maximum value of the velocity equation

The maximum value of the velocity equation is when \(\sin{(\omega t + \frac{\pi}{8})}\) is equal to 1, its maximum value: \(\sin{(\omega t + \frac{\pi}{8})} = 1\) Now we need to discover at what time the sine part becomes equal to 1.

Find the time at which the maximum velocity is attained

To find the time at which \(\sin{(\omega t + \frac{\pi}{8})} = 1\), we consider the inverse sine function with 1: \(\omega t + \frac{\pi}{8} = \sin^{-1}(1)\) Since the inverse sine of 1 is π/2 (or every odd multiple of π/2), we can write: \(\omega t + \frac{\pi}{8} = \frac{\pi (2n+1)}{4}\) , where n is an integer. Now, solve for t: \(t = \frac{1}{\omega}\left(\frac{\pi (2n+1)}{4} - \frac{\pi}{8}\right)\) \(t = \frac{3 \pi(2n+1)}{16\omega}\) By checking the options, we notice that option (C) corresponds to the case when n=0:

Choose the correct answer

The time at which the S.H.O. will attain its maximum velocity is: (C) \(\frac{3 \pi}{16 \omega}\)

Key concepts

Differentiation in Physics

Differentiation in physics is a mathematical tool used to determine rates of change. This is particularly helpful when analyzing motion, as it allows us to find how quickly one quantity changes with respect to another. In the context of simple harmonic motion (SHM), differentiation is often applied to the displacement equation to find the velocity. The displacement equation of SHM is usually given in trigonometric form, like a sine or cosine function, based on phase and angular frequency.

For example, given the displacement equation \(x = A \cos(\omega t + \phi)\), we differentiate this equation with respect to time \(t\) to find the velocity \(v\). Using the chain rule, we take the derivative of the cosine function. The chain rule aids us in applying differentiation where a function is nested within another function. By differentiating, we obtain the velocity equation:

• \(v = \frac{dx}{dt} = -A\omega \sin(\omega t + \phi)\)

This resulting equation gives us the velocity component of SHM, reflecting how fast the object is moving per unit of time. Differentiation thus serves as a crucial bridge from knowing position (through displacement) to knowing speed or velocity.

Velocity in SHM

Velocity in simple harmonic motion (SHM) describes how quickly an object oscillating back and forth is moving at any given point in time. SHM's hallmark is the constant interplay between maximum velocities and stationary points (points of zero velocity), which creates a predictable cadence of motion.

In SHM, the object speeds up as it moves toward its equilibrium position and slows down as it reaches its maximum displacements. The velocity equation derived from differentiating the displacement function is:

• \(v = -A\omega \sin(\omega t + \phi)\)

The negative sign indicates direction, primarily that the velocity is out of phase with the displacement by 90 degrees (or \(\pi/2\) radians). This equation tells us:

• The amplitude \(A\omega\) represents the maximum speed.
• The sine function \(\sin(\omega t + \phi)\) determines the current velocity's value, swinging between -1 and 1 as the object oscillates.

When \(\sin(\omega t + \phi) = 1\), we see maximum positive velocity, and when \(\sin(\omega t + \phi) = -1\), we observe the maximum negative velocity. Understanding this velocity within SHM is crucial for predicting the motion's behavior over time.

Angular Frequency in SHM

Angular frequency (\(\omega\)) in simple harmonic motion (SHM) is a critical parameter that describes how rapidly an object completes one cycle of its motion. It is directly proportional to the frequency of oscillation but expressed in radians per second, offering a more mathematical view compared to cycles per second.

The significance of angular frequency is seen in its consistent appearance within trigonometric functions that describe SHM. It marks how quickly the angle in the unit circle is subtended, effectively translating angular movement into periodic linear motion. Specifically, the general form of the displacement equation for SHM is:

• \( x = A \cos(\omega t + \phi)\)

Here, \(\omega t\) signifies the total phase angle with time, indicating how far into the oscillatory cycle the motion has progressed.

Angular frequency ties directly to essential physical properties:

• Higher angular frequency implies more cycles per second, hence faster oscillations.
• It determines the time taken for completing one full cycle (period \(T\)), where \(T = \frac{2\pi}{\omega}\).

In real-world applications involving SHM, accurately understanding and determining \(\omega\) is key to analyzing the dynamics of oscillating systems, whether in engineering, physics, or other scientific fields.