Question:
If \(y = \frac{3}{\sqrt{2}}\left(\sin \frac{\pi}{4}t + \cos \frac{\pi}{4}t\right)\) cm represents simple harmonic motion, the time period of the motion is:
If \(y = \frac{3}{\sqrt{2}}\left(\sin \frac{\pi}{4}t + \cos \frac{\pi}{4}t\right)\) cm represents simple harmonic motion, the time period of the motion is:
Show Hint
Convert sum of sine and cosine into single sine or cosine using amplitude-phase formula to identify angular frequency and period.
Updated On: Jun 19, 2026
- 8 s
- 10 s
- 6 s
- 12 s
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Express in standard SHM form.
\[ y = \frac{3}{\sqrt{2}} \left( \sin \frac{\pi}{4}t + \cos \frac{\pi}{4}t \right) = \frac{3}{\sqrt{2}} \sqrt{2} \sin\left(\frac{\pi}{4}t + \frac{\pi}{4}\right) = 3 \sin\left(\frac{\pi}{4}t + \frac{\pi}{4}\right) \]
Step 2: Identify angular frequency.
For standard SHM \(y = A \sin(\omega t + \phi)\), \(\omega = \frac{\pi}{4}\ \text{rad/s}\).
Step 3: Find period.
\[ T = \frac{2\pi}{\omega} = \frac{2\pi}{\pi/4} = 8~\text{s} \]
Step 4: Conclusion.
Hence, the time period of motion is 8 s.
\[ y = \frac{3}{\sqrt{2}} \left( \sin \frac{\pi}{4}t + \cos \frac{\pi}{4}t \right) = \frac{3}{\sqrt{2}} \sqrt{2} \sin\left(\frac{\pi}{4}t + \frac{\pi}{4}\right) = 3 \sin\left(\frac{\pi}{4}t + \frac{\pi}{4}\right) \]
Step 2: Identify angular frequency.
For standard SHM \(y = A \sin(\omega t + \phi)\), \(\omega = \frac{\pi}{4}\ \text{rad/s}\).
Step 3: Find period.
\[ T = \frac{2\pi}{\omega} = \frac{2\pi}{\pi/4} = 8~\text{s} \]
Step 4: Conclusion.
Hence, the time period of motion is 8 s.
Was this answer helpful?
0
0
Top AP EAPCET Physics Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The radii of the two columns in a U - tube are r1 and r2. When a liquid of density p (angle of contact is $ 0{}^\circ $ ) is filled in it, the level difference of the liquid in the two arms is h. The surface tension of the liquid is: (g = acceleration due to gravity)
- A rigid metallic sphere is spinning around its own axis in the absence of external torque. If the temperature is raised, its volume increases by $9 \%$. The change in its angular speed is
View More Questions
Top AP EAPCET Waves and Oscillations Questions
- In a time of 2 s, the amplitude of a damped oscillator becomes \( \frac{1}{e} \) times its initial amplitude \( A \). In the next two seconds, the amplitude of the oscillator is:
A particle is executing simple harmonic motion with a time period of 3 s. At a position where the displacement of the particle is 60% of its amplitude, the ratio of the kinetic and potential energies of the particle is:
- A body executes simple harmonic motion under the action of a force $F_1$ with a frequency $n_1$ and under another force $F_2$ with frequency $n_2$. If both forces act simultaneously in the same direction, the frequency of oscillation of the body is
- A pendulum of time period one second is losing its mechanical energy due to damping. Its mechanical energy at time $ t = 0 $ is 45 J. After completing 15 oscillations, its mechanical energy is 15 J. The ratio of the damping constant and the mass of the object making damped oscillations is:
- Which of the following statements regarding the damping force of a damped oscillator is NOT correct? Options:
View More Questions
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is
View More Questions