What is the difference of the effects of a force and a torque exerted on an object?

Answer 1

When a force F (bold characters indicate vectors) and a point defined by a position vector r (the vector from the reference system origin to the given point) the torque (T) is defined as

T = r X F (1)

where X represents vector product. In principle a torque can be defined whatever force and point are given, but it is really useful concept when rotations are involved. For example, if we consider a material point rotating around a rotation center individuated by the vector r under the action of a suitable force Fstarting from the second Newton principle

F = m a (2)

where a is the body acceleration and m is its mass, the following equation can be derived

T = J dw/dt (3)

where the torque is calculated with respect to the center of rotation, w is the angular velocity, that is a vector that is perpendicular to the instantaneous rotation plane and has an intensity equal to the time derivative of the rotation angle and J is the so called inertia momentum, that is given by the body mass multiplied by the square of the distance from the instantaneous rotation axis (thus it depends only on the body and not on the force).

Equation (3) is completely equivalent to equation (2) and it can be used to calculate the motion characteristics, like velocity, trajectory and so on. In case of substantially rotational motion, solving equation (3) can be quite simple that solving equation (2).

Moreover this is a first example of generalized motion equation, that prelude to the methods of analytical mechanics.