Advertisements
Advertisements
प्रश्न
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?
Advertisements
उत्तर
\[\text{Let } \overrightarrow{f_1},\overrightarrow{f_2},\overrightarrow{f_3},....\overrightarrow{f_n}\text{ be the forces acting on a point P.}\]
Let O be the point along which torques \(moments) will be taken.
Let:-
\[ \overrightarrow{f_1} + \overrightarrow{f_2} + \overrightarrow{f_3} + . . . + \overrightarrow{f_n} = \overrightarrow{R}..............(1)\]
Moments of force (torque) \[\overrightarrow{f_i}\] about O will be
\[ \overrightarrow{\tau_1} = \overrightarrow{OP} \times \overrightarrow{f_1} \]
The sum of the torques about O will be
\[ \overrightarrow{M} = \overrightarrow{OP} \times \overrightarrow{f_1} + \overrightarrow{OP} \times \overrightarrow{f_2} + . . . + \overrightarrow{OP} \times \overrightarrow{f_n} \]
\[ \Rightarrow \overrightarrow{M} = \overrightarrow{OP} \times \left( \overrightarrow{f_1} + \overrightarrow{f_2} + \overrightarrow{f_3} + . . . + \overrightarrow{f_n} \right)\]
\[ \Rightarrow \overrightarrow{M} = \overrightarrow{OP} \times \overrightarrow{R}...............\left[\text{From (1)}\right]\]
Thus, we see that the torque of the resultant force \[\overrightarrow{R}\] of the forces \[\overrightarrow{f_1},\overrightarrow{f_2},\overrightarrow{f_3},.....,\overrightarrow{f_n} \] gives the sum of the moments of the torques.
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.
The torque of the weight of any body about any vertical axis is zero. If it always correct?
If the resultant torque of all the forces acting on a body is zero about a point, is it necessary that it will be zero about any other point?
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 rectangular brick is kept on a table with a part of its length projecting out. It remains at rest if the length projected is slightly less than half the total length but it falls down if the length projected is slightly more than half the total length. Give reason.
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?
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 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 6⋅5 m long ladder rests against a vertical wall reaching a height of 6⋅0 m. A 60 kg man stands half way up the ladder.
- Find the torque of the force exerted by the man on the ladder about the upper end of the ladder.
- Assuming the weight of the ladder to be negligible as compared to the man and assuming the wall to be smooth, find the force exerted by the ground on the ladder.
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, ______
Define torque and mention its unit.
What are the conditions in which force can not produce torque?
A Merry-go-round, made of a ring-like platform of radius R and mass M, is revolving with angular speed ω. A person of mass M is standing on it. At one instant, the person jumps off the round, radially away from the centre of the round (as seen from the round). The speed of the round afterwards 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.
A uniform cube of mass m and side a is placed on a frictionless horizontal surface. A vertical force F is applied to the edge as shown in figure. Match the following (most appropriate choice):
| (a) mg/4 < F < mg/2 | (i) Cube will move up. |
| (b) F > mg/2 | (ii) Cube will not exhibit motion. |
| (c) F > mg | (iii) Cube will begin to rotate and slip at A. |
| (d) F = mg/4 | (iv) Normal reaction effectively at a/3 from A, no motion. |
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']
