Dynamics and inclined planes

In a nutshell

To solve problems where there is motion on an inclined plane, you can use the same principles already seen in static equilibrium problems. However any motion in a particular plane can be considered by using $$F=ma$$ in a particular direction. You can still use $$\sum F=0$$​ for a plane where there is no motion as well as $$F=\mu R$$.

Equations

Variable definitions

Dynamics and inclined planes

Procedure

Example 1

A fox of mass $$m$$ slides from rest on a slide inclined at $$\theta = 52^{\circ}$$ to the horizontal. The fox accelerates down the slide at $$a=2.4 \ m.s^{-2}$$. Calculate the coefficient of friction $$\mu .$$​

First draw a force diagram:

Resolve the forces into components, parallel and perpendicular to the plane. You should add labels to your diagram as follows:

Resolving perpendicular to the plane gives:

$$R=W \cos(52) \\ R=mg \cos(52)$$​​

Resolving parallel to the plane gives:

$$\begin{aligned}F &=ma \\P = W \sin(52) - F &= ma \\mg \sin(52) - \mu \times mg \cos(52) &= m \times 2.4 \\\end{aligned}$$​​

Solve for $$\mu$$ to give:

$$\begin{aligned}\mu &= \dfrac{mg \sin(52)-2.4m}{mg \cos(52)} \\\\\mu&= \underline{0.882 \ (3 \ s.f.)}\end{aligned}$$​​