A tunnel is dug in the earth across one of its diameter. Two masses $‘m’\,\& \,‘2m’$ are dropped from the ends of the tunnel. The masses collide and stick to each other and perform $S.H.M.$ Then amplitude of $S.H.M.$ will be : [$R =$ radius of the earth]
Advanced
Download our app for free and get started
$2 m u-m u=3 m v$
$\Rightarrow v=u / 3 ; \frac{1}{2} m u^{2}=\frac{3 G M m}{2 R}-\frac{G M m}{R}$
$u=\sqrt{\frac{G m}{R}}$
$\Delta U(r)=+\frac{G M \times 3 m}{2 R^{3}} A^{2}=\frac{1}{2} \times 3 m \cdot v^{2}$
or $\frac{G M}{2 R^{3}} A^{2}=\frac{1}{2} \cdot \frac{1}{9} \cdot \frac{G M}{R}$
$\therefore \quad \frac{A^{2}}{R^{2}}=\frac{1}{9} \Rightarrow A=\frac{R}{3}$
Download our appand get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Similar Questions
- 1A mass $m$ attached to a spring oscillates every $2\, sec$. If the mass is increased by $2 \,kg$, then time-period increases by $1\, sec$. The initial mass is ..... $kg$View Solution
- 2A $3\ kg$ sphere dropped through air has a terminal speed of $25\ m/s$. (Assume that the drag force is $-bv$.) Now suppose the sphere is attached to a spring of force constant $k = 300\ N/m$, and that it oscillates with an initial amplitude of $20\ cm$. What is the angular frequencu of its damped $SHM$? ..... $rad/s$View Solution
- 3A particle performs simple harmonic motion with amplitude A. Its speed is increased to three times at an instant when its displacement is $\frac{2 \mathrm{~A}}{3}$. The new amplitude of motion is $\frac{\mathrm{nA}}{3}$. The value of $\mathrm{n}$ is____.View Solution
- 4A simple harmonic oscillator has an amplitude $A$ and time period $6 \pi$ second. Assuming the oscillation starts from its mean position, the time required by it to travel from $x=A$ to $x=\frac{\sqrt{3}}{2} A$ will be $\frac{\pi}{x}$ s, where $x=$__________.View Solution
- 5silver atom in a solid oscillates in simple harmonic motion in some direction with a frequency of $10^{12} /sec$. What is the force constant of the bonds connecting one atom with the other? ................ $\mathrm{N/m}$ (Mole wt. of silver $= 108 $ andAvagadro number $= 6.02 \times 10^{23}$ $gm \ mole^{ -1}$ )View Solution
- 6A spring of force constant $k$ is cut into lengths of ratio $1:2:3$ . They are connected in series and the new force constant is $k'$ . Then they are connected in parallel and force constant is $k''$ . Then $k':k''$ isView Solution
- 7Column $I$ describe some situations in which a small object moves. Column $II$ describes some characteristics of these motions. Match the situation in Column $I$ with the characteristics in Column $II$ and indicate your answer by darkening appropriate bubbles in the $4 \times 4$ matrix given in the $ORS$.View Solution
Column $I$ Column $II$ $(A)$ The object moves on the $\mathrm{x}$-axis under a conservative force in such a way that its "speed" and "position" satisfy $v=c_1 \sqrt{c_2-x^2}$, where $\mathrm{c}_1$ and $\mathrm{c}_2$ are positive constants. $(p)$ The object executes a simple harmonic motion. $(B)$ The object moves on the $\mathrm{x}$-axis in such a way that its velocity and its displacement from the origin satisfy $\mathrm{v}=-\mathrm{kx}$, where $\mathrm{k}$ is a positive constant. $(q)$ The object does not change its direction. $(C)$ The object is attached to one end of a massless spring of a given spring constant. The other end of the spring is attached to the ceiling of an elevator. Initially everything is at rest. The elevator starts going upwards with a constant acceleration a. The motion of the object is observed from the elevator during the period it maintains this acceleration. $(r)$ The kinetic energy of the object keeps on decreasing. $(D)$ The object is projected from the earth's surface vertically upwards with a speed $2 \sqrt{\mathrm{GM}_e / R_e}$, where, $M_e$ is the mass of the earth and $R_e$ is the radius of the earth. Neglect forces from objects other than the earth. $(s)$ The object can change its direction only once. - 8A particle executes simple harmonic motion with a frequency $f$. The frequency with which its kinetic energy oscillates isView Solution
- 9The displacement equation of a particle is $x = 3\sin 2t + 4\cos 2t.$ The amplitude and maximum velocity will be respectivelyView Solution
- 10A pendulum is suspended in a lift and its period of oscillation when the lift is stationary is $T_0$. What must be the acceleration of the lift for the period of oscillation of the pendulum to be $T_0/2$ ?View Solution