140398
The angular velocity and the amplitude of a simple pendulum is $\omega$ and $A$ respectively. At a displacement $x$ from the mean position, if its kinetic energy is $T$ and potential energy is $U$, then the ratio of $T$ to $U$ is
C For a simple pendulum, expression for kinetic energy $(\mathrm{KE})$ and potential energy $(\mathrm{PE})$ is- $\mathrm{KE}=\mathrm{T}=\frac{1}{2} \mathrm{~m} \omega^{2}\left(\mathrm{~A}^{2}-\mathrm{x}^{2}\right)$ $\mathrm{PE}=\mathrm{U}=\frac{1}{2} \mathrm{~m} \omega^{2} \mathrm{x}^{2}$ Then the ratio of $\mathrm{PE}$ and K.E, $\frac{T}{U}=\frac{\frac{1}{2} m \omega^{2}\left(A^{2}-x^{2}\right)}{\frac{1}{2} m \omega^{2} x^{2}}$ $\frac{T}{U}=\frac{A^{2}-x^{2}}{x^{2}}$
AP EAMCET (Medical)-07.10.2020
Oscillations
140399
In $\mathrm{SHM}$ restoring force is $\mathrm{F}=-\mathrm{kx}$, where $\mathrm{k}$ is force constant, $x$ is displacement and $A$ is amplitude of motion, then total energy depends upon
1 $\mathrm{k}, \mathrm{A}$ and $\mathrm{m}$
2 $\mathrm{k}, \mathrm{x}, \mathrm{m}$
3 $\mathrm{k}, \mathrm{A}$
4 $\mathrm{k}, \mathrm{x}$
Explanation:
C We know that, K.E. $=\frac{1}{2} m \omega^{2}\left(A^{2}-x^{2}\right)$ $\text { P.E. }=\frac{1}{2} \mathrm{kx}^{2}$ $\text { P.E. }=\frac{1}{2} \mathrm{~m}^{2} \mathrm{x}^{2} \quad\left[\therefore \omega=\sqrt{\frac{\mathrm{k}}{\mathrm{m}}}\right]$ In S.H.M, total energy $=$ Potential energy + Kinetic energy $\text { T.E. }=\frac{1}{2} \mathrm{~m} \omega^{2} \mathrm{x}^{2}+\frac{1}{2} \mathrm{~m} \omega^{2}\left(\mathrm{~A}^{2}-\mathrm{x}^{2}\right)$ $\text { T.E. }=\frac{1}{2} \mathrm{~m} \omega^{2} \mathrm{~A}^{2}$ $\text { T.E. }=\frac{1}{2} \mathrm{kA} \mathrm{A}^{2}$ Where, $\mathrm{k}=$ Force constant $=\mathrm{m} \omega^{2}$ From this we can say that, total energy depends on $\mathrm{k}$ and a.
AIPMT-2001
Oscillations
140400
The oscillation of a body on a smooth horizontal surface is represented by the equation, $X=A \cos (\omega t)$ Where, $X=$ displacement at time $t$ $\omega=\text { frequency of oscillation }$ Which one of the following graphs shows correctly the variation a with $t$ ?
1
2
3
4 Here, $\mathbf{a}=$ acceleration at time $t$ $\mathbf{T}=$ Time period
Explanation:
C Given, $\mathrm{x}=\mathrm{A} \cos \omega \mathrm{t}$ $\therefore \operatorname{Velocity}(\mathrm{v})=\frac{\mathrm{dx}}{\mathrm{dt}}=\frac{\mathrm{d}}{\mathrm{dt}}(\mathrm{A} \cos \omega \mathrm{t})$ $\mathrm{v}=-\mathrm{A} \omega \sin \omega \mathrm{t}$ Acceleration $(\mathrm{a})=\frac{\mathrm{dv}}{\mathrm{dt}}=\frac{\mathrm{d}}{\mathrm{dt}}(-\mathrm{A} \omega \sin \omega \mathrm{t})$ $\mathrm{a}=-\mathrm{A} \omega^{2} \cos \omega \mathrm{t}$ Hence, graph given in option (c) shows the correct variation a with $\mathrm{t}$.
AIPMT-2014
Oscillations
140401
The total mechanical energy of a harmonic oscillator of $A=1 \mathrm{~m}$ and force constant 200 $\mathrm{Nm}^{-1}$ is $150 \mathrm{~J}$. Then
1 the minimum PE is zero
2 the minimum PE is $100 \mathrm{~J}$
3 the minimum PE is $50 \mathrm{~J}$
4 the maximum $\mathrm{KE}$ is $150 \mathrm{~J}$
Explanation:
C Given, force constant $(\mathrm{K})=200 \mathrm{~N} / \mathrm{m}$, amplitude $(\mathrm{A})=1 \mathrm{~m}, \mathrm{~T} . \mathrm{E} .=150 \mathrm{~J}$ $\because \quad$ Total energy $=$ mechanical energy T.E. $=$ K. $E_{\max }+$ P. $E_{\min }$ $\text { K. } E_{\text {max }}=\frac{1}{2} \mathrm{KA}^{2}=\frac{1}{2} \times 200 \times 1^{2}=100 \mathrm{~J}$ $150=100+\mathrm{P} . \mathrm{E}_{\min }$ $\text { or } \quad \text { P. } E_{\min }=50 \mathrm{~J}$.
AP EAMCET-1991
Oscillations
140404
A body is executing S.H.M. Its potential energy is $E_{1}$ and $E_{2}$ at displacement $x$ and $y$ respectively. The potential energy at displacement $(x+y)$ is
D The body is executing S.H.M. Its potential energy $E_{1}$ and $E_{2}$ at displacement $x$ and $y$ respectively the potential energy at displacement $(x+y)$ is Potential energy at $\mathrm{x}=\mathrm{E}_{1}=\frac{1}{2} \mathrm{kx}^{2}$ $\mathrm{x}=\sqrt{\frac{2 \mathrm{E}_{1}}{\mathrm{k}}}$ Potential energy at $\mathrm{y}=\mathrm{E}_{2}=\frac{1}{2} \mathrm{ky}^{2}$ $\mathrm{y}=\sqrt{\frac{2 \mathrm{E}_{2}}{\mathrm{k}}}$ Potential energy at $\mathrm{x}+\mathrm{y}=\mathrm{E}=\frac{1}{2} \mathrm{k}(\mathrm{x}+\mathrm{y})^{2}$ $\mathrm{E}=\frac{1}{2} \mathrm{k}\left(\sqrt{\frac{2 \mathrm{E}_{1}}{\mathrm{k}}}+\sqrt{\frac{2 \mathrm{E}_{2}}{\mathrm{k}}}\right)^{2}$ $\sqrt{\mathrm{E}}=\sqrt{\mathrm{E}_{1}}+\sqrt{\mathrm{E}_{2}}$
140398
The angular velocity and the amplitude of a simple pendulum is $\omega$ and $A$ respectively. At a displacement $x$ from the mean position, if its kinetic energy is $T$ and potential energy is $U$, then the ratio of $T$ to $U$ is
C For a simple pendulum, expression for kinetic energy $(\mathrm{KE})$ and potential energy $(\mathrm{PE})$ is- $\mathrm{KE}=\mathrm{T}=\frac{1}{2} \mathrm{~m} \omega^{2}\left(\mathrm{~A}^{2}-\mathrm{x}^{2}\right)$ $\mathrm{PE}=\mathrm{U}=\frac{1}{2} \mathrm{~m} \omega^{2} \mathrm{x}^{2}$ Then the ratio of $\mathrm{PE}$ and K.E, $\frac{T}{U}=\frac{\frac{1}{2} m \omega^{2}\left(A^{2}-x^{2}\right)}{\frac{1}{2} m \omega^{2} x^{2}}$ $\frac{T}{U}=\frac{A^{2}-x^{2}}{x^{2}}$
AP EAMCET (Medical)-07.10.2020
Oscillations
140399
In $\mathrm{SHM}$ restoring force is $\mathrm{F}=-\mathrm{kx}$, where $\mathrm{k}$ is force constant, $x$ is displacement and $A$ is amplitude of motion, then total energy depends upon
1 $\mathrm{k}, \mathrm{A}$ and $\mathrm{m}$
2 $\mathrm{k}, \mathrm{x}, \mathrm{m}$
3 $\mathrm{k}, \mathrm{A}$
4 $\mathrm{k}, \mathrm{x}$
Explanation:
C We know that, K.E. $=\frac{1}{2} m \omega^{2}\left(A^{2}-x^{2}\right)$ $\text { P.E. }=\frac{1}{2} \mathrm{kx}^{2}$ $\text { P.E. }=\frac{1}{2} \mathrm{~m}^{2} \mathrm{x}^{2} \quad\left[\therefore \omega=\sqrt{\frac{\mathrm{k}}{\mathrm{m}}}\right]$ In S.H.M, total energy $=$ Potential energy + Kinetic energy $\text { T.E. }=\frac{1}{2} \mathrm{~m} \omega^{2} \mathrm{x}^{2}+\frac{1}{2} \mathrm{~m} \omega^{2}\left(\mathrm{~A}^{2}-\mathrm{x}^{2}\right)$ $\text { T.E. }=\frac{1}{2} \mathrm{~m} \omega^{2} \mathrm{~A}^{2}$ $\text { T.E. }=\frac{1}{2} \mathrm{kA} \mathrm{A}^{2}$ Where, $\mathrm{k}=$ Force constant $=\mathrm{m} \omega^{2}$ From this we can say that, total energy depends on $\mathrm{k}$ and a.
AIPMT-2001
Oscillations
140400
The oscillation of a body on a smooth horizontal surface is represented by the equation, $X=A \cos (\omega t)$ Where, $X=$ displacement at time $t$ $\omega=\text { frequency of oscillation }$ Which one of the following graphs shows correctly the variation a with $t$ ?
1
2
3
4 Here, $\mathbf{a}=$ acceleration at time $t$ $\mathbf{T}=$ Time period
Explanation:
C Given, $\mathrm{x}=\mathrm{A} \cos \omega \mathrm{t}$ $\therefore \operatorname{Velocity}(\mathrm{v})=\frac{\mathrm{dx}}{\mathrm{dt}}=\frac{\mathrm{d}}{\mathrm{dt}}(\mathrm{A} \cos \omega \mathrm{t})$ $\mathrm{v}=-\mathrm{A} \omega \sin \omega \mathrm{t}$ Acceleration $(\mathrm{a})=\frac{\mathrm{dv}}{\mathrm{dt}}=\frac{\mathrm{d}}{\mathrm{dt}}(-\mathrm{A} \omega \sin \omega \mathrm{t})$ $\mathrm{a}=-\mathrm{A} \omega^{2} \cos \omega \mathrm{t}$ Hence, graph given in option (c) shows the correct variation a with $\mathrm{t}$.
AIPMT-2014
Oscillations
140401
The total mechanical energy of a harmonic oscillator of $A=1 \mathrm{~m}$ and force constant 200 $\mathrm{Nm}^{-1}$ is $150 \mathrm{~J}$. Then
1 the minimum PE is zero
2 the minimum PE is $100 \mathrm{~J}$
3 the minimum PE is $50 \mathrm{~J}$
4 the maximum $\mathrm{KE}$ is $150 \mathrm{~J}$
Explanation:
C Given, force constant $(\mathrm{K})=200 \mathrm{~N} / \mathrm{m}$, amplitude $(\mathrm{A})=1 \mathrm{~m}, \mathrm{~T} . \mathrm{E} .=150 \mathrm{~J}$ $\because \quad$ Total energy $=$ mechanical energy T.E. $=$ K. $E_{\max }+$ P. $E_{\min }$ $\text { K. } E_{\text {max }}=\frac{1}{2} \mathrm{KA}^{2}=\frac{1}{2} \times 200 \times 1^{2}=100 \mathrm{~J}$ $150=100+\mathrm{P} . \mathrm{E}_{\min }$ $\text { or } \quad \text { P. } E_{\min }=50 \mathrm{~J}$.
AP EAMCET-1991
Oscillations
140404
A body is executing S.H.M. Its potential energy is $E_{1}$ and $E_{2}$ at displacement $x$ and $y$ respectively. The potential energy at displacement $(x+y)$ is
D The body is executing S.H.M. Its potential energy $E_{1}$ and $E_{2}$ at displacement $x$ and $y$ respectively the potential energy at displacement $(x+y)$ is Potential energy at $\mathrm{x}=\mathrm{E}_{1}=\frac{1}{2} \mathrm{kx}^{2}$ $\mathrm{x}=\sqrt{\frac{2 \mathrm{E}_{1}}{\mathrm{k}}}$ Potential energy at $\mathrm{y}=\mathrm{E}_{2}=\frac{1}{2} \mathrm{ky}^{2}$ $\mathrm{y}=\sqrt{\frac{2 \mathrm{E}_{2}}{\mathrm{k}}}$ Potential energy at $\mathrm{x}+\mathrm{y}=\mathrm{E}=\frac{1}{2} \mathrm{k}(\mathrm{x}+\mathrm{y})^{2}$ $\mathrm{E}=\frac{1}{2} \mathrm{k}\left(\sqrt{\frac{2 \mathrm{E}_{1}}{\mathrm{k}}}+\sqrt{\frac{2 \mathrm{E}_{2}}{\mathrm{k}}}\right)^{2}$ $\sqrt{\mathrm{E}}=\sqrt{\mathrm{E}_{1}}+\sqrt{\mathrm{E}_{2}}$
NEET Test Series from KOTA - 10 Papers In MS WORD
WhatsApp Here
Oscillations
140398
The angular velocity and the amplitude of a simple pendulum is $\omega$ and $A$ respectively. At a displacement $x$ from the mean position, if its kinetic energy is $T$ and potential energy is $U$, then the ratio of $T$ to $U$ is
C For a simple pendulum, expression for kinetic energy $(\mathrm{KE})$ and potential energy $(\mathrm{PE})$ is- $\mathrm{KE}=\mathrm{T}=\frac{1}{2} \mathrm{~m} \omega^{2}\left(\mathrm{~A}^{2}-\mathrm{x}^{2}\right)$ $\mathrm{PE}=\mathrm{U}=\frac{1}{2} \mathrm{~m} \omega^{2} \mathrm{x}^{2}$ Then the ratio of $\mathrm{PE}$ and K.E, $\frac{T}{U}=\frac{\frac{1}{2} m \omega^{2}\left(A^{2}-x^{2}\right)}{\frac{1}{2} m \omega^{2} x^{2}}$ $\frac{T}{U}=\frac{A^{2}-x^{2}}{x^{2}}$
AP EAMCET (Medical)-07.10.2020
Oscillations
140399
In $\mathrm{SHM}$ restoring force is $\mathrm{F}=-\mathrm{kx}$, where $\mathrm{k}$ is force constant, $x$ is displacement and $A$ is amplitude of motion, then total energy depends upon
1 $\mathrm{k}, \mathrm{A}$ and $\mathrm{m}$
2 $\mathrm{k}, \mathrm{x}, \mathrm{m}$
3 $\mathrm{k}, \mathrm{A}$
4 $\mathrm{k}, \mathrm{x}$
Explanation:
C We know that, K.E. $=\frac{1}{2} m \omega^{2}\left(A^{2}-x^{2}\right)$ $\text { P.E. }=\frac{1}{2} \mathrm{kx}^{2}$ $\text { P.E. }=\frac{1}{2} \mathrm{~m}^{2} \mathrm{x}^{2} \quad\left[\therefore \omega=\sqrt{\frac{\mathrm{k}}{\mathrm{m}}}\right]$ In S.H.M, total energy $=$ Potential energy + Kinetic energy $\text { T.E. }=\frac{1}{2} \mathrm{~m} \omega^{2} \mathrm{x}^{2}+\frac{1}{2} \mathrm{~m} \omega^{2}\left(\mathrm{~A}^{2}-\mathrm{x}^{2}\right)$ $\text { T.E. }=\frac{1}{2} \mathrm{~m} \omega^{2} \mathrm{~A}^{2}$ $\text { T.E. }=\frac{1}{2} \mathrm{kA} \mathrm{A}^{2}$ Where, $\mathrm{k}=$ Force constant $=\mathrm{m} \omega^{2}$ From this we can say that, total energy depends on $\mathrm{k}$ and a.
AIPMT-2001
Oscillations
140400
The oscillation of a body on a smooth horizontal surface is represented by the equation, $X=A \cos (\omega t)$ Where, $X=$ displacement at time $t$ $\omega=\text { frequency of oscillation }$ Which one of the following graphs shows correctly the variation a with $t$ ?
1
2
3
4 Here, $\mathbf{a}=$ acceleration at time $t$ $\mathbf{T}=$ Time period
Explanation:
C Given, $\mathrm{x}=\mathrm{A} \cos \omega \mathrm{t}$ $\therefore \operatorname{Velocity}(\mathrm{v})=\frac{\mathrm{dx}}{\mathrm{dt}}=\frac{\mathrm{d}}{\mathrm{dt}}(\mathrm{A} \cos \omega \mathrm{t})$ $\mathrm{v}=-\mathrm{A} \omega \sin \omega \mathrm{t}$ Acceleration $(\mathrm{a})=\frac{\mathrm{dv}}{\mathrm{dt}}=\frac{\mathrm{d}}{\mathrm{dt}}(-\mathrm{A} \omega \sin \omega \mathrm{t})$ $\mathrm{a}=-\mathrm{A} \omega^{2} \cos \omega \mathrm{t}$ Hence, graph given in option (c) shows the correct variation a with $\mathrm{t}$.
AIPMT-2014
Oscillations
140401
The total mechanical energy of a harmonic oscillator of $A=1 \mathrm{~m}$ and force constant 200 $\mathrm{Nm}^{-1}$ is $150 \mathrm{~J}$. Then
1 the minimum PE is zero
2 the minimum PE is $100 \mathrm{~J}$
3 the minimum PE is $50 \mathrm{~J}$
4 the maximum $\mathrm{KE}$ is $150 \mathrm{~J}$
Explanation:
C Given, force constant $(\mathrm{K})=200 \mathrm{~N} / \mathrm{m}$, amplitude $(\mathrm{A})=1 \mathrm{~m}, \mathrm{~T} . \mathrm{E} .=150 \mathrm{~J}$ $\because \quad$ Total energy $=$ mechanical energy T.E. $=$ K. $E_{\max }+$ P. $E_{\min }$ $\text { K. } E_{\text {max }}=\frac{1}{2} \mathrm{KA}^{2}=\frac{1}{2} \times 200 \times 1^{2}=100 \mathrm{~J}$ $150=100+\mathrm{P} . \mathrm{E}_{\min }$ $\text { or } \quad \text { P. } E_{\min }=50 \mathrm{~J}$.
AP EAMCET-1991
Oscillations
140404
A body is executing S.H.M. Its potential energy is $E_{1}$ and $E_{2}$ at displacement $x$ and $y$ respectively. The potential energy at displacement $(x+y)$ is
D The body is executing S.H.M. Its potential energy $E_{1}$ and $E_{2}$ at displacement $x$ and $y$ respectively the potential energy at displacement $(x+y)$ is Potential energy at $\mathrm{x}=\mathrm{E}_{1}=\frac{1}{2} \mathrm{kx}^{2}$ $\mathrm{x}=\sqrt{\frac{2 \mathrm{E}_{1}}{\mathrm{k}}}$ Potential energy at $\mathrm{y}=\mathrm{E}_{2}=\frac{1}{2} \mathrm{ky}^{2}$ $\mathrm{y}=\sqrt{\frac{2 \mathrm{E}_{2}}{\mathrm{k}}}$ Potential energy at $\mathrm{x}+\mathrm{y}=\mathrm{E}=\frac{1}{2} \mathrm{k}(\mathrm{x}+\mathrm{y})^{2}$ $\mathrm{E}=\frac{1}{2} \mathrm{k}\left(\sqrt{\frac{2 \mathrm{E}_{1}}{\mathrm{k}}}+\sqrt{\frac{2 \mathrm{E}_{2}}{\mathrm{k}}}\right)^{2}$ $\sqrt{\mathrm{E}}=\sqrt{\mathrm{E}_{1}}+\sqrt{\mathrm{E}_{2}}$
140398
The angular velocity and the amplitude of a simple pendulum is $\omega$ and $A$ respectively. At a displacement $x$ from the mean position, if its kinetic energy is $T$ and potential energy is $U$, then the ratio of $T$ to $U$ is
C For a simple pendulum, expression for kinetic energy $(\mathrm{KE})$ and potential energy $(\mathrm{PE})$ is- $\mathrm{KE}=\mathrm{T}=\frac{1}{2} \mathrm{~m} \omega^{2}\left(\mathrm{~A}^{2}-\mathrm{x}^{2}\right)$ $\mathrm{PE}=\mathrm{U}=\frac{1}{2} \mathrm{~m} \omega^{2} \mathrm{x}^{2}$ Then the ratio of $\mathrm{PE}$ and K.E, $\frac{T}{U}=\frac{\frac{1}{2} m \omega^{2}\left(A^{2}-x^{2}\right)}{\frac{1}{2} m \omega^{2} x^{2}}$ $\frac{T}{U}=\frac{A^{2}-x^{2}}{x^{2}}$
AP EAMCET (Medical)-07.10.2020
Oscillations
140399
In $\mathrm{SHM}$ restoring force is $\mathrm{F}=-\mathrm{kx}$, where $\mathrm{k}$ is force constant, $x$ is displacement and $A$ is amplitude of motion, then total energy depends upon
1 $\mathrm{k}, \mathrm{A}$ and $\mathrm{m}$
2 $\mathrm{k}, \mathrm{x}, \mathrm{m}$
3 $\mathrm{k}, \mathrm{A}$
4 $\mathrm{k}, \mathrm{x}$
Explanation:
C We know that, K.E. $=\frac{1}{2} m \omega^{2}\left(A^{2}-x^{2}\right)$ $\text { P.E. }=\frac{1}{2} \mathrm{kx}^{2}$ $\text { P.E. }=\frac{1}{2} \mathrm{~m}^{2} \mathrm{x}^{2} \quad\left[\therefore \omega=\sqrt{\frac{\mathrm{k}}{\mathrm{m}}}\right]$ In S.H.M, total energy $=$ Potential energy + Kinetic energy $\text { T.E. }=\frac{1}{2} \mathrm{~m} \omega^{2} \mathrm{x}^{2}+\frac{1}{2} \mathrm{~m} \omega^{2}\left(\mathrm{~A}^{2}-\mathrm{x}^{2}\right)$ $\text { T.E. }=\frac{1}{2} \mathrm{~m} \omega^{2} \mathrm{~A}^{2}$ $\text { T.E. }=\frac{1}{2} \mathrm{kA} \mathrm{A}^{2}$ Where, $\mathrm{k}=$ Force constant $=\mathrm{m} \omega^{2}$ From this we can say that, total energy depends on $\mathrm{k}$ and a.
AIPMT-2001
Oscillations
140400
The oscillation of a body on a smooth horizontal surface is represented by the equation, $X=A \cos (\omega t)$ Where, $X=$ displacement at time $t$ $\omega=\text { frequency of oscillation }$ Which one of the following graphs shows correctly the variation a with $t$ ?
1
2
3
4 Here, $\mathbf{a}=$ acceleration at time $t$ $\mathbf{T}=$ Time period
Explanation:
C Given, $\mathrm{x}=\mathrm{A} \cos \omega \mathrm{t}$ $\therefore \operatorname{Velocity}(\mathrm{v})=\frac{\mathrm{dx}}{\mathrm{dt}}=\frac{\mathrm{d}}{\mathrm{dt}}(\mathrm{A} \cos \omega \mathrm{t})$ $\mathrm{v}=-\mathrm{A} \omega \sin \omega \mathrm{t}$ Acceleration $(\mathrm{a})=\frac{\mathrm{dv}}{\mathrm{dt}}=\frac{\mathrm{d}}{\mathrm{dt}}(-\mathrm{A} \omega \sin \omega \mathrm{t})$ $\mathrm{a}=-\mathrm{A} \omega^{2} \cos \omega \mathrm{t}$ Hence, graph given in option (c) shows the correct variation a with $\mathrm{t}$.
AIPMT-2014
Oscillations
140401
The total mechanical energy of a harmonic oscillator of $A=1 \mathrm{~m}$ and force constant 200 $\mathrm{Nm}^{-1}$ is $150 \mathrm{~J}$. Then
1 the minimum PE is zero
2 the minimum PE is $100 \mathrm{~J}$
3 the minimum PE is $50 \mathrm{~J}$
4 the maximum $\mathrm{KE}$ is $150 \mathrm{~J}$
Explanation:
C Given, force constant $(\mathrm{K})=200 \mathrm{~N} / \mathrm{m}$, amplitude $(\mathrm{A})=1 \mathrm{~m}, \mathrm{~T} . \mathrm{E} .=150 \mathrm{~J}$ $\because \quad$ Total energy $=$ mechanical energy T.E. $=$ K. $E_{\max }+$ P. $E_{\min }$ $\text { K. } E_{\text {max }}=\frac{1}{2} \mathrm{KA}^{2}=\frac{1}{2} \times 200 \times 1^{2}=100 \mathrm{~J}$ $150=100+\mathrm{P} . \mathrm{E}_{\min }$ $\text { or } \quad \text { P. } E_{\min }=50 \mathrm{~J}$.
AP EAMCET-1991
Oscillations
140404
A body is executing S.H.M. Its potential energy is $E_{1}$ and $E_{2}$ at displacement $x$ and $y$ respectively. The potential energy at displacement $(x+y)$ is
D The body is executing S.H.M. Its potential energy $E_{1}$ and $E_{2}$ at displacement $x$ and $y$ respectively the potential energy at displacement $(x+y)$ is Potential energy at $\mathrm{x}=\mathrm{E}_{1}=\frac{1}{2} \mathrm{kx}^{2}$ $\mathrm{x}=\sqrt{\frac{2 \mathrm{E}_{1}}{\mathrm{k}}}$ Potential energy at $\mathrm{y}=\mathrm{E}_{2}=\frac{1}{2} \mathrm{ky}^{2}$ $\mathrm{y}=\sqrt{\frac{2 \mathrm{E}_{2}}{\mathrm{k}}}$ Potential energy at $\mathrm{x}+\mathrm{y}=\mathrm{E}=\frac{1}{2} \mathrm{k}(\mathrm{x}+\mathrm{y})^{2}$ $\mathrm{E}=\frac{1}{2} \mathrm{k}\left(\sqrt{\frac{2 \mathrm{E}_{1}}{\mathrm{k}}}+\sqrt{\frac{2 \mathrm{E}_{2}}{\mathrm{k}}}\right)^{2}$ $\sqrt{\mathrm{E}}=\sqrt{\mathrm{E}_{1}}+\sqrt{\mathrm{E}_{2}}$
140398
The angular velocity and the amplitude of a simple pendulum is $\omega$ and $A$ respectively. At a displacement $x$ from the mean position, if its kinetic energy is $T$ and potential energy is $U$, then the ratio of $T$ to $U$ is
C For a simple pendulum, expression for kinetic energy $(\mathrm{KE})$ and potential energy $(\mathrm{PE})$ is- $\mathrm{KE}=\mathrm{T}=\frac{1}{2} \mathrm{~m} \omega^{2}\left(\mathrm{~A}^{2}-\mathrm{x}^{2}\right)$ $\mathrm{PE}=\mathrm{U}=\frac{1}{2} \mathrm{~m} \omega^{2} \mathrm{x}^{2}$ Then the ratio of $\mathrm{PE}$ and K.E, $\frac{T}{U}=\frac{\frac{1}{2} m \omega^{2}\left(A^{2}-x^{2}\right)}{\frac{1}{2} m \omega^{2} x^{2}}$ $\frac{T}{U}=\frac{A^{2}-x^{2}}{x^{2}}$
AP EAMCET (Medical)-07.10.2020
Oscillations
140399
In $\mathrm{SHM}$ restoring force is $\mathrm{F}=-\mathrm{kx}$, where $\mathrm{k}$ is force constant, $x$ is displacement and $A$ is amplitude of motion, then total energy depends upon
1 $\mathrm{k}, \mathrm{A}$ and $\mathrm{m}$
2 $\mathrm{k}, \mathrm{x}, \mathrm{m}$
3 $\mathrm{k}, \mathrm{A}$
4 $\mathrm{k}, \mathrm{x}$
Explanation:
C We know that, K.E. $=\frac{1}{2} m \omega^{2}\left(A^{2}-x^{2}\right)$ $\text { P.E. }=\frac{1}{2} \mathrm{kx}^{2}$ $\text { P.E. }=\frac{1}{2} \mathrm{~m}^{2} \mathrm{x}^{2} \quad\left[\therefore \omega=\sqrt{\frac{\mathrm{k}}{\mathrm{m}}}\right]$ In S.H.M, total energy $=$ Potential energy + Kinetic energy $\text { T.E. }=\frac{1}{2} \mathrm{~m} \omega^{2} \mathrm{x}^{2}+\frac{1}{2} \mathrm{~m} \omega^{2}\left(\mathrm{~A}^{2}-\mathrm{x}^{2}\right)$ $\text { T.E. }=\frac{1}{2} \mathrm{~m} \omega^{2} \mathrm{~A}^{2}$ $\text { T.E. }=\frac{1}{2} \mathrm{kA} \mathrm{A}^{2}$ Where, $\mathrm{k}=$ Force constant $=\mathrm{m} \omega^{2}$ From this we can say that, total energy depends on $\mathrm{k}$ and a.
AIPMT-2001
Oscillations
140400
The oscillation of a body on a smooth horizontal surface is represented by the equation, $X=A \cos (\omega t)$ Where, $X=$ displacement at time $t$ $\omega=\text { frequency of oscillation }$ Which one of the following graphs shows correctly the variation a with $t$ ?
1
2
3
4 Here, $\mathbf{a}=$ acceleration at time $t$ $\mathbf{T}=$ Time period
Explanation:
C Given, $\mathrm{x}=\mathrm{A} \cos \omega \mathrm{t}$ $\therefore \operatorname{Velocity}(\mathrm{v})=\frac{\mathrm{dx}}{\mathrm{dt}}=\frac{\mathrm{d}}{\mathrm{dt}}(\mathrm{A} \cos \omega \mathrm{t})$ $\mathrm{v}=-\mathrm{A} \omega \sin \omega \mathrm{t}$ Acceleration $(\mathrm{a})=\frac{\mathrm{dv}}{\mathrm{dt}}=\frac{\mathrm{d}}{\mathrm{dt}}(-\mathrm{A} \omega \sin \omega \mathrm{t})$ $\mathrm{a}=-\mathrm{A} \omega^{2} \cos \omega \mathrm{t}$ Hence, graph given in option (c) shows the correct variation a with $\mathrm{t}$.
AIPMT-2014
Oscillations
140401
The total mechanical energy of a harmonic oscillator of $A=1 \mathrm{~m}$ and force constant 200 $\mathrm{Nm}^{-1}$ is $150 \mathrm{~J}$. Then
1 the minimum PE is zero
2 the minimum PE is $100 \mathrm{~J}$
3 the minimum PE is $50 \mathrm{~J}$
4 the maximum $\mathrm{KE}$ is $150 \mathrm{~J}$
Explanation:
C Given, force constant $(\mathrm{K})=200 \mathrm{~N} / \mathrm{m}$, amplitude $(\mathrm{A})=1 \mathrm{~m}, \mathrm{~T} . \mathrm{E} .=150 \mathrm{~J}$ $\because \quad$ Total energy $=$ mechanical energy T.E. $=$ K. $E_{\max }+$ P. $E_{\min }$ $\text { K. } E_{\text {max }}=\frac{1}{2} \mathrm{KA}^{2}=\frac{1}{2} \times 200 \times 1^{2}=100 \mathrm{~J}$ $150=100+\mathrm{P} . \mathrm{E}_{\min }$ $\text { or } \quad \text { P. } E_{\min }=50 \mathrm{~J}$.
AP EAMCET-1991
Oscillations
140404
A body is executing S.H.M. Its potential energy is $E_{1}$ and $E_{2}$ at displacement $x$ and $y$ respectively. The potential energy at displacement $(x+y)$ is
D The body is executing S.H.M. Its potential energy $E_{1}$ and $E_{2}$ at displacement $x$ and $y$ respectively the potential energy at displacement $(x+y)$ is Potential energy at $\mathrm{x}=\mathrm{E}_{1}=\frac{1}{2} \mathrm{kx}^{2}$ $\mathrm{x}=\sqrt{\frac{2 \mathrm{E}_{1}}{\mathrm{k}}}$ Potential energy at $\mathrm{y}=\mathrm{E}_{2}=\frac{1}{2} \mathrm{ky}^{2}$ $\mathrm{y}=\sqrt{\frac{2 \mathrm{E}_{2}}{\mathrm{k}}}$ Potential energy at $\mathrm{x}+\mathrm{y}=\mathrm{E}=\frac{1}{2} \mathrm{k}(\mathrm{x}+\mathrm{y})^{2}$ $\mathrm{E}=\frac{1}{2} \mathrm{k}\left(\sqrt{\frac{2 \mathrm{E}_{1}}{\mathrm{k}}}+\sqrt{\frac{2 \mathrm{E}_{2}}{\mathrm{k}}}\right)^{2}$ $\sqrt{\mathrm{E}}=\sqrt{\mathrm{E}_{1}}+\sqrt{\mathrm{E}_{2}}$