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?
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.
Explain why friction is necessary to make the disc in Figure roll in the direction indicated
(a) Give the direction of frictional force at B, and the sense of frictional torque, before perfect rolling begins.
(b) What is the force of friction after perfect rolling begins?
The torque of the weight of any body about any vertical axis is zero. If it always correct?
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 heavy particle of mass m falls freely near the earth's surface. What is the torque acting on this particle about a point 50 cm east to the line of motion? Does this torque produce any angular acceleration in the particle?
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
The density of a rod gradually decreases from one end to the other. It is pivoted at an end so that it can move about a vertical axis though the pivot. A horizontal force F is applied on the free end in a direction perpendicular to the rod. The quantities, that do not depend on which end of the rod is pivoted, are ________________ .
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?
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?
Calculate the total torque acting on the body shown in the following figure about the point O.
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 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, ______
What are the conditions in which force can not produce torque?
A particle of mass 5 units is moving with a uniform speed of v = `3sqrt 2` units in the XOY plane along the line y = x + 4. Find the magnitude of angular momentum
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. |
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 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 ______.
