Two blocks $A$ and $B$ each of mass m are connected by a massless spring of natural length L and spring constant $K$. The blocks are initially resting on a smooth horizontal floor with the spring at its natural length as shown in figure. A third identical block $C$ also of mass $m$ moves on the floor with a speed $v$ along the line joining $A$ and $B$ and collides with $A$. Then
AThe kinetic energy of the $A-B$ system at maximum compression of the spring is zero
BThe kinetic energy of the $ A-B$ system at maximum compression of the spring is $\frac{m{v^2}}{4}$
CThe maximum compression of the spring is $v\sqrt {m/2K} $
D
Both (b) and (c)
IIT 1993, Diffcult
Download our app for free and get started
D
Both (b) and (c)
d (d) Let the velocity acquired by $A$ and $B$ be $V$, then
$mv = mV + mV $
$\Rightarrow V = \frac{v}{2}$
Also $\frac{1}{2}m{v^2} = \frac{1}{2}m{V^2} + \frac{1}{2}m{V^2} + \frac{1}{2}k{x^2}$
Where $ x$ is the maximum compression of the spring. On solving the above equations, we get $x = v{\left( {\frac{m}{{2k}}} \right)^{1/2}}$
At maximum compression, kinetic energy of the
$A -B$ system $ = \frac{1}{2}m{V^2} + \frac{1}{2}m{V^2} = m{V^2} = \frac{{m{v^2}}}{4}$
Download our app
and 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.*
A point particle is acted upon by a restoring force $-k x^{3}$. The time-period of oscillation is $T$, when the amplitude is $A$. The time-period for an amplitude $2 A$ will be
A particle excutes $SHM$ on a straight line path. The amplitude of oscillation is $2\,cm$. When the displacement of the particle from the mean position is $1\,cm$, the numerical value of magnitude of acceleration is equal to the numerical value of magnitude of velocity. The frequency of $SHM$ is (in $second^{-1}$)
A mass $m$ attached to a spring oscillates with a period of $3\,s$. If the mass is increased by $1\,kg$ the period increases by $1\,s$. The initial mass $m$ is
Two small bodies of mass of $2\, kg$ each attached to each other using a thread of length $10\, cm$, hang on a spring whose force constant is $200\, N/m$, as shown in the figure. We burn the thread. What is the distance between the two bodies when the top body first arrives at its highest position .... $cm$ ? (Take $\pi^2 = 10$)
A trolley of mass $m_1$ is placed on horizontal rigid pair of rails at same height. A mass $m_2$ is suspended to the trolley vertically by me ans of a ideal massless rope. The rope hangs between rails without touching them. Trolley can move along smooth rails but can't move in any other direction. Suspended mass is given small oscillation and perform $SHM$ after displacing small from stable equilibrium position in two ways, first perpendicular to the rails and second parallel to the rails. The ratio of time period of these (second case to first case) two $SHM's$ is
Time period of a simple pendulum is $T$ inside a lift when the lift is stationary. If the lift moves upwards with an acceleration $g / 2,$ the time period of pendulum will be
Two pendulums have time periods $T$ and $5T/4.$ They start $SHM$ at the same time from the mean position. After how many oscillations of the smaller pendulum they will be again in the same phase :
A uniform stick of mass $M$ and length $L$ is pivoted at its centre. Its ends are tied to two springs each of force constant $K$ . In the position shown in figure, the strings are in their natural length. When the stick is displaced through a small angle $\theta $ and released. The stick