Torque has dimensions of energy

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Torque is fundamentally different from energy, as it is a vector while energy is a scalar. The relationship between torque and energy is established through the concept of work, where rotational work is defined as torque multiplied by the angle of rotation. Angles are measured in radians, which are dimensionless, meaning torque must encompass all dimensions of work itself. Consequently, torque can be expressed in SI units as "Newton metres per radian," highlighting its similarity to work. This establishes torque as the rotational counterpart to work done in linear motion.
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I just noticed that torque has dimensions of energy! What does that mean?
 
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It doesn't really mean anything. Energy is a scalar and torque is a vector, so they are fundamentally different quantities. In the same way that, for translational energy, work is the force (or, rather, the component of force in the direction of motion) times distance moved, for rotational energy work is the torque times the angle rotated through. We conventionally measure angles in radians, which are dimensionless since they are ratios of distances. Thus, torque must carry all the dimensions of work by itself. If you like, you can consider the SI dimensions of torque to "really" be "Newton metres per radian". It's the same idea as angular frequency which mathematically has units of "1/seconds" but really represents the dimensionally equivalent units of "radians per second".
 
It means torque also has the same units as work. It is the rotational analogue of work done in a translational manner.
 
Thanks a lot..
 

Thread 'Derivation of Hamilton's Principle: Questions'

I have recently been really interested in the derivation of Hamiltons Principle. On my research I found that with the term ##m \cdot \frac{d}{dt} (\frac{dr}{dt} \cdot \delta r) = 0## (1) one may derivate ##\delta \int (T - V) dt = 0## (2). The derivation itself I understood quiet good, but what I don't understand is where the equation (1) came from, because in my research it was just given and not derived from anywhere. Does anybody know where (1) comes from or why from it the...
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