Torque the vector cross product, p is the particle's linear momentum, and r is the displacement vector from the origin (the origin is assumed to be a fixed location anywhere in space). 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 formula_17 (whether or not mass is constant) and the definition of velocity formula_18 The cross product of momentum formula_20 with its associated velocity formula_21 is zero because velocity and momentum are parallel, so the second term vanishes. By definition, torque τ
Brain-to-body mass ratio animals usually have larger brains than smaller animals); the relationship is not, however, linear. Small mammals such as mice may have a brain/body ratio similar to humans, while elephants have a comparatively lower brain/body ratio. In animals, it is thought that the larger the brain, the more brain weight will be available for more complex cognitive tasks. However, large animals need more neurons to represent their own bodies and control specific muscles; thus, relative rather than absolute brain size makes for a ranking of animals that better coincides with the observed complexity of animal behaviour. The relationship between brain-to-body mass