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I’m taking Introductions to Physics book. In the book on the chapter "Physics & Mathematics" there was one question where we were supposed to find the magnitude of work done from given vectors Force & Displacement.

Using Dot product I successfully calculated the number part of Magnitude and wrote Nₘ as unit.

Then I looked at the answer and it was written: 4.0N−m or 4.0J.

I understand Joules but how is we have - instead of multiplication?

We can’t add difrent quantities, is that why we use - as multiplication operator. If yes why did we even needed it? What purpose does it justify?

Answer 1

The notation 'N-m' symbolizes the multiplication of newtons by meters to measure work, which equals a joule (J). Work performed by a force during displacement can be positive or negative, indicating energy transfer to or from the system, and is calculated using the dot product of force and displacement vectors.

Step-by-step explanation:

The confusion lies in the notation used for the unit of work in physics - the hyphen used in '4.0N-m' is not intended to represent subtraction but rather multiplication or a product of force and displacement. The correct unit for work is newton-meter (N·m), which is equivalent to a joule (J). One joule is the amount of work done when a force of one newton displaces an object by one meter in the direction of the force.

When a force exerted on a system has a component in the direction of motion, work is done, and energy is transferred. For instance, pulling a briefcase across a distance entails positive work being done on the briefcase. Conversely, if the force acts in the opposite direction of the movement, negative work is done, as energy is removed from the system.

The calculation of work using dot product involves multiplying the force components by the displacement components in their respective directions: W = Fx Dx + Fy Dy + Fz Dz. Thus, work performed by a force, which results in the displacement of an object, has both magnitude and direction, and can be analyzed as vectors.