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
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.
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.
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?
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?
When a force of 6⋅0 N is exerted at 30° to a wrench at a distance of 8 cm from the nut it is just able to loosen the nut. What force F would be sufficient to loosen it if it acts perpendicularly to the wrench at 16 cm from the nut?
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.
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.
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 ______.
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 ______.
Choose the correct alternatives:
- For a general rotational motion, angular momentum L and angular velocity ω need not be parallel.
- For a rotational motion about a fixed axis, angular momentum L and angular velocity ω are always parallel.
- For a general translational motion , momentum p and velocity v are always parallel.
- For a general translational motion, acceleration a and velocity v are always parallel.
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.
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 door is hinged at one end and is free to rotate about a vertical axis (Figure). Does its weight cause any torque about this axis? Give reason for your answer.
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 magnitude of the torque on a particle of mass 1 kg is 2.5 Nm about the origin. If the force acting on it is 1 N, and the distance of the particle from the origin is 5 m, the angle between the force and the position vector is (in radians) ______.
