Question

A particle of mass ‘m’ is executing oscillations about the origin on the x-axis. Its potential energy is U(x) = K|x|³ where K is a positive constant. If the amplitude of oscillation is ‘a’ then its time period T is –

• A. Proportional to $\frac{1}{\sqrt{a}}$
• B. Independent of a
• C. Proportional to $\sqrt{a}$
• D. Proportional to a^3/2

Answer 1

Answer: (A)

Proportional to $\frac{1}{\sqrt{a}}$

Answer 2

Let time period depends on mass (m), amplitude (1) & constant (k) as –

$T \propto m^{\alpha}a^{\beta}k^{\gamma}$

$\left\lbrack \frac{ML^{2}T^{- 2}}{L^{3}} \right\rbrack^{\gamma}$

$M^{\alpha + \gamma}L^{\beta - \gamma +}T^{- 2\gamma}$

Equating dimensions both sides

a + g = 0 ̃ a = –g a = 1/2

b – g = 0 ̃ b = g b = –1/2

–2g = 1 ̃ g = –1/2

$t \propto m^{1/2}a^{- 1/2}k^{1/2}$

t µ $\sqrt{\frac{m}{ak}}$

t µ $\frac{1}{\sqrt{a}}$.