Work and toque are both products of force and distance. Are work and torque the same?

Answer 1

Work is the vector product of force and distance (not scalar). Force and distance need to be in the same direction, otherwise you take only the component of the force which is in the same direction to compute work.

Torque is a scalar product of force and radius. Imagine a jar of jam with the lid stuck by dried sugar. You have to exert quite a lot of force to open it. But if you could multiply the outer diameter of the lid by 10, you could open the jar by expanding 1/10 of that force. Note though that you would have to exert this reduced force over a longer distance (exactly 10 time longer) so that the net work would be the same.

Answer 2

No, work and torque are NOT the same:

In the case of WORK, it's the FORCE times the DISTANCE MOVED IN THE DIRECTION OF THE FORCE that matters.

In the case of a TORQUE, you have the STATIC moment of a FORCE multiplied by a DISTANCE (the "lever arm") PERPENDICULAR TO THE FORCE.

In vector terms, work involves a scalar product, while torque involves a vector product.

[ LATER NOTE: I'm afraid that in his answer immediately below, 'catarthur' employs the terms "vector product" or "scalar product" COMPLETELY THE WRONG WAY ROUND. Work done is fully equivalent to the energy expended, and so naturally has the same dimensions as energy. That means it involves a SCALAR product (otherwise called the "dot product," in vector language) of force applied and distance moved. This is consistent with the fact that energy per se has NO vector character. THAT is why it's a SCALAR.

A torque IS a vector product (also called the "cross product" in vector language) because the result IS still a vector. This is PHYSICALLY OBVIOUS : a torque has both MAGNITUDE --- its "strength" --- AND a DIRECTION associated with it, the axis or direction around which it operates and could POTENTIALLY turn something. So, it's away from the computer screen and back to hitting the books for a lot of clearly needed revision, for you, 'catarthur' ! ]

A TORQUE only does work if it MOVES (usually, but not always ROTATES) something. (A torque CAN be applied to ultimately move something in a linear direction.) The WORK done is once again the APPLIED FORCE multiplied by the DISTANCE MOVED IN THE DIRECTION OF THE FORCE. In the simplest case when the force applied is perpendicular to a STRAIGHT lever arm, the latter "distance moved" is the length of the lever arm TIMES the angle theta (in radians) through which it was moved. The final work then done by the torque is the force TIMES the length of the lever arm TIMES the angle through which it was moved.

The work done by a torque has the same DIMENSIONS as work done by a linearly directed force because both end products involve the magnitude of a force component and a length multiplying that component. The additional angle factor used in the case of the torque doesn't affect the end product's dimensions, because the angle moved is itself dimensionless. (Think "radians" defined by a circular arc length divided by a radial length.)

In the case of work from both a (linearly directed) force, and from a (rotational) torque, the work done involves the component of force ALONG the direction of motion. In the case of a torque, the physical nature and shape of the lever itself is NOT important; it could wander from the axis to the point of the force's application in the most meandering and fantastical way. In the end, the work done by a torque is the product of the radius vector from the rotational axis to the point of application, and the force component along the direction perpendicular to that radius vector.

[ Note that the force applied need not necessarily be applied at the physical place that you'd like to rotate. For example, a very long wrench makes it easier to turn a small, otherwise "frozen" nut because the long lever arm can "magnify" the effect of a given modest force applied at its end. This magnifying "principle of the lever" was enunciated by Archimedes, of course. ]

Live long and prosper.

Answer 3

Not the same; work is done over a period of time. Torque is instantaneous.