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
Find the components along the x, y, z axes of the angular momentum l of a particle, whose position vector is r with components x, y, z and momentum is p with components px, py and 'p_z`. Show that if the particle moves only in the x-y plane the angular momentum has only a z-component.
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?
If several forces act on a particle, the total torque on the particle may be obtained by first finding the resultant force and then taking torque of this resultant. Prove this. Is this result valid for the forces acting on different particles of a body in such a way that their lines of action intersect at a common point?
When a body is weighed on an ordinary balance we demand that the arum should be horizontal if the weights on the two pans are equal. Suppose equal weights are put on the two pans, the arm is kept at an angle with the horizontal and released. Is the torque of the two weights about the middle point (point of support) zero? Is the total torque zero? If so, why does the arm rotate and finally become horizontal?
A cubical block of mass m and edge a slides down a rough inclined plane of inclination θ with a uniform speed. Find the torque of the normal force acting on the block about its centre.
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?
The ratio of the acceleration for a solid sphere (mass m and radius R) rolling down an incline of angle θ without slipping and slipping down the incline without rolling is, ______
Define torque and mention its unit.
State conservation of angular momentum.
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 ______.
Figure shows two identical particles 1 and 2, each of mass m, moving in opposite directions with same speed v along parallel lines. At a particular instant, r1 and r2 are their respective position vectors drawn from point A which is in the plane of the parallel lines. Choose the correct options:
- Angular momentum l1 of particle 1 about A is l1 = mvd1
- Angular momentum l2 of particle 2 about A is l2 = mvr2
- Total angular momentum of the system about A is l = mv(r1 + r2)
- Total angular momentum of the system about A is l = mv (d2 − d1)
⊗ represents a unit vector coming out of the page.
⊗ represents a unit vector going into the page.
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.
Two discs of moments of inertia I1 and I2 about their respective axes (normal to the disc and passing through the centre), and rotating with angular speed ω2 and ω2 are brought into contact face to face with their axes of rotation coincident.
- Does the law of conservation of angular momentum apply to the situation? why?
- Find the angular speed of the two-disc system.
- Calculate the loss in kinetic energy of the system in the process.
- Account for this loss.
A spherical shell of 1 kg mass and radius R is rolling with angular speed ω on horizontal plane (as shown in figure). The magnitude of angular momentum of the shell about the origin O is `a/3 R^2` ω. The value of a will be:
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']
A particle of mass ‘m’ is moving in time ‘t’ on a trajectory given by
`vecr = 10alphat^2hati + 5beta(t - 5)hatj`
Where α and β are dimensional constants.
The angular momentum of the particle becomes the same as it was for t = 0 at time t = ______ seconds.
Angular momentum of a single particle moving with constant speed along the circular path ______.
