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The work done is given by the dot product of the two vectors, where the result is a scalar. When the force F is constant and the angle θ between the force and the displacement s is also constant, then the work done is given by: = If the force is variable, then work is given by the line integral:
Power is the rate with respect to time at which work is done; it is the time derivative of work: =, where P is power, W is work, and t is time. We will now show that the mechanical power generated by a force F on a body moving at the velocity v can be expressed as the product: P = d W d t = F ⋅ v {\displaystyle P={\frac {dW}{dt}}=\mathbf {F ...
Work: W: Transferred energy joule (J) L 2 M T −2: scalar Young's modulus: E: Ratio of stress to strain pascal (Pa = N/m 2) L −1 M T −2: scalar; assumes isotropic linear material spring constant: k: k is the torsional constant (measured in N·m/radian), which characterizes the stiffness of the torsional spring or the resistance to angular ...
Lagrange wrote in his Mécanique analytique (published 1788, based on work done around 1755) that mechanics can be viewed as operating in a four-dimensional space— three dimensions of space, and one of time. [4] As early as 1827, Möbius realized that a fourth spatial dimension would allow a three-dimensional form to be rotated onto its ...
The first term in D'Alembert's principle above is the virtual work done by the non-constraint forces N k along the virtual displacements δr k, and can without loss of generality be converted into the generalized analogues by the definition of generalized forces = =, so that = = = = = =.
The dimension of a vector space is the number of vectors in any basis for the space, i.e. the number of coordinates necessary to specify any vector. This notion of dimension (the cardinality of a basis) is often referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension.
Enjoy a classic game of Hearts and watch out for the Queen of Spades!
the integral of force with respect to the distance (s) the object has travelled ( , work) has dimension T −2 L 2 M. In economics, one distinguishes between stocks and flows : a stock has a unit (say, widgets or dollars), while a flow is a derivative of a stock, and has a unit of the form of this unit divided by one of time (say, dollars ...