The displacement of a particle is SHM varies according to the relation x = 4 (cos πt + sin πt). The amplitude of the particle is 

• − 4

• 4

• $4\sqrt{2}$

• 8

What is the phase difference between two simple harmonic motions represented by ${\mathrm{x}}_1 = \mathrm{A} \sin \left(\mathrm{\omega t} + \frac{\mathrm{\pi }}{6}\right)$ and x₂ = A cos (ωt) ?

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{2\mathrm{\pi }}{3}$

When a spring is stretched by 10 cm, the potential energy stored is E. When the spring is stretched by 10 cm more, the potential energy stored in the spring becomes

• 2 E

• 4 E

• 6 E

• 10 E

When a particle executing SHM oscillates with a frequency v, then the kinetic energy of the particle

• changes periodically with a frequency of v

• changes periodically with a frequency of 2v

• changes periodically with a frequency of v/2

• remains constant

The displacement of a particle in a periodic motion is given by $\mathrm{y} = 4 {\cos }^2 \left(\frac{\mathrm{t}}{2}\right) \sin \left(1000 \mathrm{t}\right)$. This displacement may be considered as the result of superposition of n independent harmonic oscillations. Here n is

• 1

• 2

• 3

• 4

A simple pendulum of length L swings in a vertical plane. The tension of the string when it makes an angle θ with the vertical and the bob of mass in moves with a speed v is (g is the gravitational acceleration)

• mv²/L

• mg cos θ + mv² / L

• mg cos θ − mv² / L

• mg cos θ

257.

The displacement x of a particle varies with time t as, $\mathrm{x} = {\mathrm{ae}}^{-\mathrm{\alpha t}} +\mathrm{b} {\mathrm{e}}^{ \mathrm{\beta t}}$ where a, b α and β are positive constants. The velocity of the particle will

• go on decreasing with time

• be independent of $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$

• drop to zero when $\mathrm{\alpha } = \mathrm{\beta }$

• go on increasing with time