Physics with Calculus/Appendix 2/Examples of Derivatives

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< Physics with Calculus

Motion[edit]

For x(t), position as a function of time

Velocity: The rate of change of position with respect to time

\begin{matrix}
\mathbf{v}(t) = f'(t) = {dx \over dt}
\end{matrix}

Acceleration: The rate of change of velocity with respect to time

\begin{matrix}
\mathbf{a}(t) = \mathbf{v}'(t) = f''(t) = {d^2x \over dt^2}
\end{matrix}

Jerk: The rate of change of acceleration with respect to time

\begin{matrix}
\mathbf{j}(t) = \mathbf{a}'(t) = f'''(t) = {d^3x \over dt^3}
\end{matrix}

Jerk is not commonly used in first year motion. Its main application is in dealing with travel of large objects that change their weight as they move due to a change in mass. One example might be a rocket travelling up from rest. As it burns fuel, its centre of gravity changes and as such, its acceleration is not constant (violation of Newton's Second Law).

Mechanics (Statics)[edit]

Given the details of the loading of a beam, we can represent it on a diagram of the beam with arrows indicating forces, curved arrows indicating moments (resistance to torque) and shaded regions representing universally varying or distributed loads. We can use this diagram (commonly known as a free body diagram) and the information contained within it to draw a diagram representing the shear forces (V in the beam, and can also derive an equation that represents these. The equation may not be as simple as a polynomial, and is quite often a series of continuous functions with endpoints at the points on the beam where the forces occur.

We can perform an indefinite integral on each of these segments of the beam to get more information on it. The indefinite integrals combine to form a diagram of the bending moments in the beam. Bending moments are a special type of moment, as the beam is most likely to fail where the bending moments are at a relative extrema. By definition, any indefinite integral will contain a constant, C. In the case of the bending moment diagram, our C is merely the endpoint of the previous segment. The only exception being when we have a moment, we add or subtract its value (depending on direction).

Hence, differentiating the bending moment model will give us the shear force.

Differentiating the shear force model brings us back to our loading diagram, and differentiating that will give us the shape of the deflection of the beam under the loading.

For any bending moment model b, as a function of distance from the end of the beam, x,


\begin{matrix}
b'(x) &=& V(x)
\\
b''(x) &=& V'(x) &=& \operatorname{FBD}(x)
\\
b'''(x) &=& \operatorname{FBD}'(x) &=& f(x)
\end{matrix}

Where f(x) is a function describing the deflection of the beam.