A body of mass m is situated in a potential field U(x) = U0 (1-cos αx) when U0 and α are constants. Find the time period of small oscillations.

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Payal Gupta | Contributor-Level 10

4 months ago

This is a short answer type question as classified in NCERT Exemplar

given potential energy associated with the field

U (x)=U_o (1-cos $α x$ )

F=-dU (x)/dx

F=-d (U_o-U_ocos $α x$ )=-U_o $α$ sin $α x$

F=-U_{o
$α^{2} x$}

F $\propto - x$

Motion is SHM for small oscillatons

F=-mw²X

Mw²=U_{o
$α^{2}$}

w²= $\frac{U_{o} α^{2}}{m}$ , w= $\sqrt[]{\frac{U_{o} α^{2}}{m}}$

T= $\frac{2 π}{w}$ = $2 π \sqrt[]{\frac{m}{U_{o} α^{2}}}$

This is a short answer type question as classified in NCERT Exemplargiven potential energy associated with the fieldU (x)=Uo (1-cos <math> <mi>α</mi> <mi>x</mi> </math> )F=-dU (x)/dxF=-d (Uo-Uocos <math> <mi>α</mi> <mi>x</mi> </math> )=-Uo <math> <mi>α</mi> </math> sin <math> <mi>α</mi> <mi>x</mi> </math> F=-Uo <math> <msup> <mrow> <mrow> <mi>α</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mi>x</mi> </math> F <math> <mo>∝</mo> <mo>-</mo> <mi>x</mi> </math> Motion is SHM for small oscillatonsF=-mw2XMw2=Uo <math> <msup> <mrow> <mrow> <mi>α</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </math> w2= <math> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>U</mi> </mrow> </mrow> <mrow> <mrow> <mi>o</mi> </mrow> </mrow> </msub> <msup> <mrow> <mrow> <mi>α</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> </mfrac> </math>  , w= <math> <mroot> <mrow> <mrow> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>U</mi> </mrow> </mrow> <mrow> <mrow> <mi>o</mi> </mrow> </mrow> </msub> <msup> <mrow> <mrow> <mi>α</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> <mrow></mrow> </mroot> </math> T= <math> <mfrac> <mrow> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mi>w</mi> </mrow> </mrow> </mfrac> </math> = <math> <mn>2</mn> <mi>π</mi> <mroot> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <msub> <mrow> <mrow> <mi>U</mi> </mrow> </mrow> <mrow> <mrow> <mi>o</mi> </mrow> </mrow> </msub> <msup> <mrow> <mrow> <mi>α</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> </mfrac> </mrow> </mrow> <mrow></mrow> </mroot> </math>

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A body of mass m is situated in a potential field U(x) = U0 (1-cos αx) when U0 and α are constants. Find the time period of small oscillations.

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