Advertisements
Advertisements
प्रश्न
The following figure shows a rough track, a portion of which is in the form of a cylinder of radius R. With what minimum linear speed should a sphere of radius r be set rolling on the horizontal part so that it completely goes round the circle on the cylindrical part.
Advertisements
उत्तर
Let the sphere be thrown with velocity \[v'\] and its velocity becomes v at the top-most point.
From the free body diagram of the sphere, at the topmost point, we have
\[\frac{m v^2}{R - r} = mg\]
\[ \Rightarrow v^2 = g\left( R - r \right)\]
On applying the law of conservation of energy, we have
\[\left( \frac{1}{2}m\nu '^2 + \frac{1}{2}I\omega '^2 \right) = 2mg\left( R - r \right) + \left( \frac{1}{2}m \nu^2 + \frac{1}{2}I \omega^2 \right)\]
\[ \Rightarrow \frac{7}{10}m\nu '^2 = 2mg\left( R - r \right) + \frac{7}{10}m \nu^2 \]
\[ \Rightarrow \frac{7}{10}m\nu '^2 = 2mg\left( R - r \right) + \frac{7}{10}mg(R - r)\]
\[ \Rightarrow \frac{7}{10}\nu '^2 = \frac{27g\left( R - r \right)}{10}\]
\[ \Rightarrow \nu' = \sqrt{\frac{27}{7} g\left( R - r \right)}\]
APPEARS IN
संबंधित प्रश्न
A bob suspended from the ceiling of a car which is accelerating on a horizontal road. The bob stays at rest with respect to the car with the string making an angle θ with the vertical. The linear momentum of the bob as seen from the road is increasing with time. Is it a violation of conservation of linear momentum? If not, where is the external force changes the linear momentum?
If the total mechanical energy of a particle is zero, is its linear momentum necessarily zero? Is it necessarily nonzero?
If the linear momentum of a particle is known, can you find its kinetic energy? If the kinetic energy of a particle is know can you find its linear momentum?
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"?
When a nucleus at rest emits a beta particle, it is found that the velocities of the recoiling nucleus and the beta particle are not along the same straight line. How can this be possible in view of the principle of conservation of momentum?
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?
A bullet hits a block kept at rest on a smooth horizontal surface and gets embedded into it. Which of the following does not change?
Internal forces can change
In an elastic collision
(a) the kinetic energy remains constant
(b) the linear momentum remains constant
(c) the final kinetic energy is equal to the initial kinetic energy
(d) the final linear momentum is equal to the initial linear momentum.
A ball hits a floor and rebounds after an inelastic collision. In this case
(a) the momentum of the ball just after the collision is same as that just before the collision
(b) the mechanical energy of the ball remains the same during the collision
(c) the total momentum of the ball and the earth is conserved
(d) the total energy of the ball and the earth remains the same
A man of mass 50 kg starts moving on the earth and acquires a speed 1.8 m/s. With what speed does the earth recoil? Mass of earth = 6 × 1024 kg.
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 20 g travelling horizontally with a speed of 500 m/s passes through a wooden block of mass 10.0 kg initially at rest on a level surface. The bullet emerges with a speed of 100 m/s and the block slides 20 cm on the surface before coming to rest. Find the friction coefficient between the block and the surface (See figure).
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.
Two mass m1 and m2 are connected by a spring of spring constant k and are placed on a frictionless horizontal surface. Initially the spring is stretched through a distance x0 when the system is released from rest. Find the distance moved by the two masses before they again come to rest.
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.
The blocks shown in figure have equal masses. The surface of A is smooth but that of Bhas a friction coefficient of 0.10 with the floor. Block A is moving at a speed of 10 m/s towards B which is kept at rest. Find the distance travelled by B if (a) the collision is perfectly elastic and (b) the collision is perfectly inelastic.
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?
