A particle is moving with constant speed in a circular path. When the particle turns by an angle $90^{\circ}$, the ratio of instantaneous velocity to its average velocity is $\pi : x\sqrt{2}$. The value of $x$ will be (1) 2 (2) 5 (3) 1 (4) 7
Given below are two statements: one is labelled as Assertion $A$ and the other is labelled as Reason $R$. Assertion A: When a body is projected at an angle $45^{\circ}$, its range is maximum. Reason R: For maximum range, the value of $\sin 2\theta$ should be equal to one. In the light of the above statements, choose the correct answer from the options given below: (1) $A$ is false but $R$ is true (2) $A$ is true but $R$ is false (3) Both $A$ and $R$ are correct and $R$ is the correct explanation of $A$ (4) Both $A$ and $R$ are correct but $R$ is NOT the correct explanation of $A$
A mass $m$ is attached to two springs as shown in figure. The spring constants of two springs are $K_1$ and $K_2$. For the frictionless surface, the time period of oscillation of mass $m$ is (1) $2\pi\sqrt{\frac{m}{K_1 + K_2}}$ (2) $\frac{1}{2\pi}\sqrt{\frac{K_1 - K_2}{m}}$ (3) $2\pi\sqrt{\frac{m}{K_1 - K_2}}$ (4) $\frac{1}{2\pi}\sqrt{\frac{K_1 + K_2}{m}}$
A small block of mass 100 g is tied to a spring of spring constant $7.5\mathrm{~N~m}^{-1}$ and length 20 cm. The other end of spring is fixed at a particular point $A$. If the block moves in a circular path on a smooth horizontal surface with constant angular velocity $5\mathrm{~rad~s}^{-1}$ about point $A$, then tension in the spring is (1) 0.75 N (2) 0.25 N (3) 0.50 N (4) 1.5 N
A particle of mass 10 g moves in a straight line with retardation $2x$, where $x$ is the displacement in SI units. Its loss of kinetic energy for above displacement is $\frac{10^{-n}}{x}\mathrm{~J}$. The value of $n$ will be $\_\_\_\_$.