The torque of the wieght of any body about any vetical axis is zero. Is it always correct?

The net external torque on a system of particles about an axis is zero. Which of the following are compatible with it?

Assertion : If the earth stops rotating about its axis, the value of the weight of the body at equator will decrease. Reason : The centripetal force does not act on the body at the equator.

The vector sum of a system of non-collinear forces acting on a rigid body is given to be non-zero. If the vector sum of all the torques due to the system of forces about a certain point is found to be zero, does this mean that it is necessarily zero about any arbitrary point?

A body is free to rotate about an axis parallel to y- axis. A force of vec(F)=(3hat(i)+2hat(j)+6hat(k))N is acting on the body the position vector of whose point of applications is vec(r)=(2hat(i)-3hat(j))m . The moment of inertia of body about y- axis is 10kgm^(2) . The angular acceleration of body is :-

Assertion : For circular orbits, the law of periods is T^(2) prop r^(3) , where M is the mass of sun and r is the radius of orbit. Reason : The square of the period T of any planet about the sun is proportional to the cube of the semi-major axis a of the orbit.

Prove the result that the velocity v of translation of a rolling body (like a ring, disc, cylinder or sphere) at the bottom of an inclined plane of a height h is given by v^(2)=(2gh)/((1+k^(2)//R^(2))) using dynamical consideration (i.e. by consideration of forces and torques). Note k is the radius of gyration of the body about its symmetry axis, and R is the radius of the body. The body starts from rest at the top of the plane.

Prove that result that the velocity v of translation of a rolling body (like a ring, disc, cylinder or sphere) at the bottom of an inclined plane of a height h is given by v^(2)=(2gh)/((1+(k^(2))/(R^(2)))) using dynamical consideration (i.e. by consideration of forces and torques). Note k is the radius of gyration of the body about its symmetry axis, and R is the radius of the body. The body starts from rest at the top of the plane.

There are three forces F_(1) F_(2) and F_(3) acting on a body, all acting on a point P on the body. The body is found to move with uniform speed. (a) Show that the forces are coplanar. (b) Show that the torque acting on the body about any point due to these three forces is zero.

Find the curve for which the intercept cut off by any tangent on y-axis is proportional to the square of the ordinate of the point of tangency.

Satement-1: if there is no external torque on a body about its centre of mass, then the velocity of the center of mass remains constant. Statement-2: The linear momentum of an isolated system remains constant.