Advertisements
Advertisements
प्रश्न
A uniform sphere of mass m and radius R is placed on a rough horizontal surface (Figure). The sphere is struck horizontally at a height h from the floor. Match the following:
| Column I | Column II | |
| (a) h = R/2 | (i) | Sphere rolls without slipping with a constant velocity and no loss of energy. |
| (b) h = R | (ii) | Sphere spins clockwise, loses energy by friction. |
| (c) h = 3R/2 | (iii) | Sphere spins anti-clockwise, loses energy by friction. |
| (d) h = 7R/5 | (iv) | Sphere has only a translational motion, looses energy by friction. |
Advertisements
उत्तर
| Column I | Column II | |
| (a) h = R/2 | (iii) | Sphere spins anti-clockwise, loses energy by friction. |
| (b) h = R | (iv) | Sphere has only a translational motion, looses energy by friction. |
| (c) h = 3R/2 | (ii) | Sphere spins clockwise, loses energy by friction. |
| (d) h = 7R/5 | (i) | Sphere rolls without slipping with a constant velocity and no loss of energy. |
Explanation:
Mass of the sphere = m
Radius = R
h = height from the floor
The sphere will roll without slipping when ω = V/R
Where v is linear velocity and to is the angular velocity of the sphere.
Now, angular momentum of the sphere is about centre of mass .....[We are applying conservation of angular momentum just before and after struck.]
Then by the law of conservation of angular momentum
`mv(h - R) = I_ω`
`mv(h - R) = 2/5 mR^2 v/R`
`h - R = 2/5 R`
`h = 2/5 R +R = 7/5 R`
Therefore, the sphere rolls without slipping with a constant velocity and no loss of energy. Thus (d) - (i)
Torque due to force `F = τ = (h - R) xx F`
If τ = 0, h – R = 0 and thus h = R
In this case, the sphere will only have a translation motion and slip against the force of friction. Thus (b) - (iv)
For clockwise rotation of the sphere τ > 0
`(h - R) xx F > 0`
Or `h > R`, thus (c) - (ii)
For anti-clockwise rotation `τ < 0`
`(h - R) xx F < 0`
`h < R`,Thus (a) - (iii)
APPEARS IN
संबंधित प्रश्न
Two particles, each of mass m and speed v, travel in opposite directions along parallel lines separated by a distance d. Show that the angular momentum vector of the two particle system is the same whatever be the point about which the angular momentum is taken.
A solid cylinder of mass 20 kg rotates about its axis with angular speed 100 rad s–1. The radius of the cylinder is 0.25 m. What is the kinetic energy associated with the rotation of the cylinder? What is the magnitude of the angular momentum of the cylinder about its axis?
The torque of a force \[\overrightarrow F \] about a point is defined as \[\overrightarrow\Gamma = \overrightarrow r \times \overrightarrow F.\] Suppose \[\overrightarrow r, \overrightarrow F\] and \[\overrightarrow \Gamma\] are all nonzero. Is \[r \times \overrightarrow\Gamma || \overrightarrow F\] always true? Is it ever true?
A body is in translational equilibrium under the action of coplanar forces. If the torque of these forces is zero about a point, is it necessary that it will also be zero about any other point?
A ladder is resting with one end on a vertical wall and the other end on a horizontal floor. If it more likely to slip when a man stands near the bottom or near the top?
Equal torques act on the disc A and B of the previous problem, initially both being at rest. At a later instant, the linear speeds of a point on the rim of A and another point on the rim of B are \[\nu_A\] and \[\nu_B\] respectively. We have
A simple pendulum of length l is pulled aside to make an angle θ with the vertical. Find the magnitude of the torque of the weight ω of the bob about the point of suspension. When is the torque zero?
A flywheel of moment of inertia 5⋅0 kg-m2 is rotated at a speed of 60 rad/s. Because of the friction at the axle it comes to rest in 5⋅0 minutes. Find (a) the average torque of the friction (b) the total work done by the friction and (c) the angular momentum of the wheel 1 minute before it stops rotating.
A particle is moving with a constant velocity along a line parallel to the positive X-axis. The magnitude of its angular momentum with respect to the origin is, ______
A rope is wound around a hollow cylinder of mass 3 kg and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N?
Two discs of the same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of the disc with angular velocities ω1 and ω2. They are brought in to contact face to face coinciding with the axis of rotation. The expression for loss of energy during this process is, ______
What are the conditions in which force can not produce torque?
A particle of mass m is moving in yz-plane with a uniform velocity v with its trajectory running parallel to + ve y-axis and intersecting z-axis at z = a (Figure). The change in its angular momentum about the origin as it bounces elastically from a wall at y = constant is ______.
The net external torque on a system of particles about an axis is zero. Which of the following are compatible with it?
- The forces may be acting radially from a point on the axis.
- The forces may be acting on the axis of rotation.
- The forces may be acting parallel to the axis of rotation.
- The torque caused by some forces may be equal and opposite to that caused by other forces.
A rod of mass 'm' hinged at one end is free to rotate in a horizontal plane. A small bullet of mass m/4 travelling with speed 'u' hits the rod and attaches to it at its centre. Find the angular speed of rotation of rod just after the bullet hits the rod 3. [take length of the rod as 'l']
The position vector of 1 kg object is `vecr = (3hati - hatj)` m and its velocity `vecv = (3hati + hatk)` ms−1. The magnitude of its angular momentum is `sqrtx` Nm where x is ______.
A solid sphere is rotating in free space. If the radius of the sphere is increased while keeping the mass the same, which one of the following will not be affected?
