Advertisements
Advertisements
Question
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?
Advertisements
Solution
Yes , if the torque due to forces in translation equillibrium is zero about a point, it will be zero about other point in the plane.
Let us consider a planner lamina of some mass, acted upon by forces \[\overrightarrow{F_1}, \overrightarrow{F_2}, . . . \overrightarrow{F_i},\] etc.
Let a force \[\overrightarrow{F_i}\] act on a \[i^{th}\] particle and torque due to \[\overrightarrow{F_i}\] be zero at a point Q.
Since the body is in translation equillibrium, we have
\[\Sigma \overrightarrow{F_i} = 0 .............(1)\]
Again, torque about P is zero . Therefore, we have
\[\Sigma\left( \overrightarrow{r_{pi}} \times \overrightarrow{F_i} \right) = 0 ..............(2)\]
Now, torque about point Q will be
\[\Sigma \overrightarrow{r}_{Qi} \times \overrightarrow{F_i} \]
\[ = \Sigma\left( \overrightarrow{r}_{QP} + \overrightarrow{r}_{Pi} \right) \times \overrightarrow{F_i} ............\left[\text{From fig.} \right]\]
\[ = \Sigma\left( \overrightarrow{r}_{QP} \times \vec{F_i} + \overrightarrow{r_{Pi}} \times \overrightarrow{F_i} \right)\]
\[ = \Sigma \overrightarrow{r_{QP}} \times \overrightarrow{F_i} + \Sigma \overrightarrow{r_{Pi}} \times \overrightarrow{F_i} \]
\[ = \overrightarrow{r_{QP}} \times \Sigma \overrightarrow{F_i} + 0 ..........\left[\text{From (2)} \right]\]
\[ = \overrightarrow{r_{QP}} \times 0 .........\left[\text{From (1)} \right]\]
\[ = 0\]
Thus, \[\overrightarrow{F}\] is zero about any other point Q.
APPEARS IN
RELATED QUESTIONS
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?
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 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?
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 ________________ .
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, ______
State conservation of angular momentum.
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 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 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 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 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']
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 ______.
Angular momentum of a single particle moving with constant speed along the circular path ______.
