Advertisements
Advertisements
प्रश्न
Two blocks of masses m1 and m2 are connected by a spring of spring constant k (See figure). The block of mass m2 is given a sharp impulse so that it acquires a velocity v0 towards right. Find (a) the velocity of the centre of mass, (b) the maximum elongation that the spring will suffer.
Advertisements
उत्तर
Given,
Velocity of mass, m2 = v0
Velocity of mass, m1 = 0
(a) Velocity of centre of mass is given by,
\[v_{cm} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}\]
\[\Rightarrow v_{cm} = \frac{m_1 \times 0 + m_2 \times v_0}{m_1 + m_2}\]
\[ \Rightarrow v_{cm} = \frac{m_2 v_0}{m_1 + m_2}\]
(b) Let the maximum elongation in spring be x.
The spring attains maximum elongation when velocities of both the blocks become equal to the velocity of centre of mass.
i.e. v1 = v2 = vcm
On applying the law of conservation of energy, we can write:
Change in kinetic energy = Potential energy stored in spring
\[\Rightarrow \frac{1}{2} m_2 v_0^2 - \frac{1}{2}( m_1 + m_2 ) \left( \frac{m_2 v_0}{m_1 + m_2} \right)^2 = \frac{1}{2}k x^2 \]
\[ \Rightarrow m_2 v_0^2 \left( 1 - \frac{m_2}{m_1 + m_2} \right) = k x^2\]
`Rightarrow = V_o[(m_1m_2)/((m_1+m_2)K)]^(1/2)`
APPEARS IN
संबंधित प्रश्न
If the total mechanical energy of a particle is zero, is its linear momentum necessarily zero? Is it necessarily nonzero?
Suppose we define a quantity 'Linear momentum' as linear momentum = mass × speed.
The linear momentum of a system of particles is the sum of linear momenta of the individual particles. Can we state principle of conservation of linear momentum as "linear momentum of a system remains constant if no external force acts on it"?
In one-dimensional elastic collision of equal masses, the velocities are interchanged. Can velocities in a one-dimensional collision be interchanged if the masses are not equal?
Internal forces can change
The quantities remaining constant in a collisions are
A nucleus moving with a velocity \[\vec{v}\] emits an α-particle. Let the velocities of the α-particle and the remaining nucleus be v1 and v2 and their masses be m1 and m2.
A block moving in air breaks in two parts and the parts separate
(a) the total momentum must be conserved
(b) the total kinetic energy must be conserved
(c) the total momentum must change
(d) the total kinetic energy must change
A ball of mass 50 g moving at a speed of 2.0 m/s strikes a plane surface at an angle of incidence 45°. The ball is reflected by the plane at equal angle of reflection with the same speed. Calculate (a) the magnitude of the change in momentum of the ball (b) the change in the magnitude of the momentum of the ball.
Light in certain cases may be considered as a stream of particles called photons. Each photon has a linear momentum h/λ where h is the Planck's constant and λ is the wavelength of the light. A beam of light of wavelength λ is incident on a plane mirror at an angle of incidence θ. Calculate the change in the linear momentum of a photon as the beam is reflected by the mirror.
A 60 kg man skating with a speed of 10 m/s collides with a 40 kg skater at rest and they cling to each other. Find the loss of kinetic energy during the collision.
A block of mass 2.0 kg is moving on a frictionless horizontal surface with a velocity of 1.0 m/s (In the following figure) towards another block of equal mass kept at rest. The spring constant of the spring fixed at one end is 100 N/m. Find the maximum compression of the spring.
A bullet of mass 25 g is fired horizontally into a ballistic pendulum of mass 5.0 kg and gets embedded in it. If the centre of the pendulum rises by a distance of 10 cm, find the speed of the bullet.
Suppose the particle of the previous problem has a mass m and a speed \[\nu\] before the collision and it sticks to the rod after the collision. The rod has a mass M. (a) Find the velocity of the centre of mass C of the system constituting "the rod plus the particle". (b) Find the velocity of the particle with respect to C before the collision. (c) Find the velocity of the rod with respect to C before the collision. (d) Find the angular momentum of the particle and of the rod about the centre of mass C before the collision. (e) Find the moment of inertia of the system about the vertical axis through the centre of mass C after the collision. (f) Find the velocity of the centre of mass C and the angular velocity of the system about the centre of mass after the collision.
A sphere starts rolling down an incline of inclination θ. Find the speed of its centre when it has covered a distance l.
A thin spherical shell of radius R lying on a rough horizontal surface is hit sharply and horizontally by a cue. Where should it be hit so that the shell does not slip on the surface?
