MHT CET · Physics · Oscillations
A particle starts from mean position and performs S.H.M. with period 6 second. At what time its kinetic energy is \(50 \%\) of total energy? \(\left(\cos 45^{\circ}=\frac{1}{\sqrt{2}}\right)\)
- A \(0.75\) second
- B 1 second
- C \(0.25\) second
- D \(0.50\) second
Answer & Solution
Correct Answer
(A) \(0.75\) second
Step-by-step Solution
Detailed explanation
\(T=6 \mathrm{~sec}\)
\(\frac{1}{2} m \omega^{2}\left(A^{2}-x^{2}\right)=\frac{1}{2}\left(\frac{1}{2} m \omega^{2} A^{2}\right.)\)
\(A^{2}-x^{2}=\frac{A^{2}}{2}\)
\(\therefore x^{2}=\frac{A^{2}}{2} \Rightarrow x=\frac{A}{\sqrt{2}}\)
\(\frac{A}{\sqrt{2}}=A \sin \omega t=A \sin \frac{2 \pi}{T} t\)
\(\frac{1}{\sqrt{2}}=\sin \frac{2 \pi}{6} t=\sin \frac{\pi}{3} t\)
\(\sin \frac{\pi}{4} t=\sin \frac{\pi}{3} t\)
\(\therefore t=\frac{3}{4}=0.75 \mathrm{sec}\)
\(\frac{1}{2} m \omega^{2}\left(A^{2}-x^{2}\right)=\frac{1}{2}\left(\frac{1}{2} m \omega^{2} A^{2}\right.)\)
\(A^{2}-x^{2}=\frac{A^{2}}{2}\)
\(\therefore x^{2}=\frac{A^{2}}{2} \Rightarrow x=\frac{A}{\sqrt{2}}\)
\(\frac{A}{\sqrt{2}}=A \sin \omega t=A \sin \frac{2 \pi}{T} t\)
\(\frac{1}{\sqrt{2}}=\sin \frac{2 \pi}{6} t=\sin \frac{\pi}{3} t\)
\(\sin \frac{\pi}{4} t=\sin \frac{\pi}{3} t\)
\(\therefore t=\frac{3}{4}=0.75 \mathrm{sec}\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Physics
- The thermal conductivity of a rod is \(2 .\) What is its thermal resistivity?MHT CET 2007 Hard
- A solid sphere of mass \(M\), radius \(R\) has moment of inertia ' \(I\) ' about its diameter. It is recast into a disc of thickness ' \(t\) ' whose moment of inertia about an axis passing through its edge and perpendicular to its plane remains ' \(I\) '. Radius of the disc will beMHT CET 2021 Medium
- A galvanometer has resistance \(80 \Omega\) and it is shunted with resistance \(20 \Omega\). If \(20 \%\) of the main current flows through galvanometer, then what is the value of main current?MHT CET 2024 Easy
- The three vectors \(\vec{A}=3 \hat{i}-2 \hat{j}+\hat{k}, \vec{B}=\hat{i}-3 \hat{j}+5 \hat{k}\) and \(\vec{C}=2 \hat{i}-\hat{j}+4 \hat{k}\) will formMHT CET 2025 Medium
- A particle of mass \(5 \mathrm{~g}\) is executing S.H.M. with an amplitude \(0.3 \mathrm{~m}\) and period \(\frac{\pi}{5} \mathrm{~s}\). The maximum value of the force acting on the particle isMHT CET 2021 Easy
- An alternating voltage is represented by \(\mathrm{V}=80 \sin (100 \pi \mathrm{t})\) \(\cos (100 \pi \mathrm{t})\) volt. The peak voltage isMHT CET 2021 Easy
More PYQs from MHT CET
- The conductivity of 0.02 M KCl , solution is \(0.00250 \mathrm{ohm}^{-1} \mathrm{~cm}^{-1}\). What is its molar conductivity?MHT CET 2024 Easy
- Identify the correct sequence of path of water across the root.MHT CET 2024 Easy
- Identify the product formed in the following reactions,
MHT CET 2021 Medium - Which among the following salt solution in water shows \(\mathrm{pH}\) less than 7 ?MHT CET 2021 Easy
- Let two non-collinear unit vectors \(\hat{a}\) and \(\hat{b}\) form an acute angle. A point P moves, so that at any time \(t\) the position vector \(\overline{\mathrm{OP}}\), where O is the origin, is given by \(\hat{a} \cos t+\hat{b} \sin t\). When \(P\) is farthest from origin O , let M be the length of \(\overline{\mathrm{OP}}\) and \(\hat{\mathrm{u}}\) be the unit vector along \(\overline{\mathrm{OP}}\), thenMHT CET 2024 Hard
- Identify the correct order of thermal stability of hydrides of 16 group elements from the following.MHT CET 2025 Easy