valid for any type of trajectory. Note that although force and acceleration are always parallel and directly proportional, the torque τ need not be parallel or directly proportional to the angular acceleration α. This arises from the fact that although mass is always conserved, the moment of inertia in general is not. In some simple cases like a rotating disc, the moment of inertia is a constant, the rotational Newton's second law can be where and . Proof of the equivalence of definitions The definition of angular momentum for a single point particle is: where p is the particle's linear momentum and r is the position vector from the origin. The time-derivative of this is: This result can easily be proven by splitting the vectors into components and applying the product rule. Now using the definition of force (whether or not mass is constant) and the definition of velocity The cross product of momentum with its associated velocity is zero because velocity and momentum are parallel, so the second term vanishes. By definition, torque τ = r × F. Therefore, torque on a particle is equal to the first derivative of its angular momentum with respect to time. If multiple forces are applied, Newton's second law instead reads , and it follows that This is a general proof for point particles. The proof can be generalized to a system of point particles by applying the above proof to each of the point particles and then summing over all the point particles. Similarly, the proof can be generalized to a continuous mass by applying the above proof to each point within the mass, and then integrating over the entire mass. Units Torque has the dimension of force times distance, symbolically . Although those fundamental dimensions are the same as that for energy or work, official SI literature suggests using the unit newton-metre (N⋅m) and never the joule. The unit newton-metre is properly denoted N⋅m. The traditional imperial and U.S. customary units for torque are the pound foot (lbf-ft), or for small values the pound inch (lbf-in). In the US, torque is most commonly referred to as the foot-pound (denoted as either lb-ft or ft-lb) and the inch-pound (denoted as in-lb). Practitioners depend on context and the hyphen in the abbreviation to know that these refer to torque and not to energy or moment of mass (as the symbolism ft-lb would properly