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Find the magnitude of the acceleration of a box placed on a frictionless ramp that is inclined at θ = 30° above the horizontal.

We begin with a free body diagram and choice of a coordinate system. Although, as always, the choice of coordinates is arbitrary, a clever choice can help us reduce the work involved in solving the problem. Consider the two possibilities below:

Looking only at the forces in the free body diagrams, it is not clear that the tilted coordinates should be preferred. If we use the tilted coordinates, then the normal force has very simple components, N_x = 0 and N_y = +N. Gravity, however, is neither pure x or pure y in the tilted coordinates. Instead, we have mg_x = mgsinθ and mg_y = – mgcosθ. In the regular (untilted) coordinates, the situation is reversed. Gravity is simple, with mg_x = 0 and mg_y = -mg, but now the normal force is not aligned with either axis and has components N_x = Nsinθ and N_y = Ncosθ. Thus, it would seem that either choice results in the same level of simplification in the vector equations of Newton's 2nd Law.

The tiebreaker in this case is only clear if you use your physical intution about the constraints in this problem. By thinking about this problem, we know with certainty that the box will be sliding down the ramp. It will not rise off the ramp or sink down through it. Thus, the acceleration of the box will have to be parallel to the ramp. It is important to remember that the acceleration will appear in the equations of Newton's 2nd Law as well. Thus, the equations will be simpler if the acceleration has simple vector components. Clearly, the acceleration will be simpler in the tilted coordinates, having a_y = 0 and a_x = a. Thus, by using the tilted coordinates, we simplify two vectors, the normal force and the acceleration, while using untilted coordinates only simplifies gravity.

To demonstrate the utility of the tilted coordinates, we will solve the problem using both options.

Method 1 – Regular (untilted) Coordinates

Before writing Newton's 2nd Law for each direction, we must break up the acceleration.

We can then write the equations:

\begin{large}\[ \sum F_{x} = N\sin\theta = ma_{x} = ma\cos\theta \]
\[ \sum F_{y} = N\cos\theta - mg = ma_{y} = -ma\sin\theta \]\end{large}

We can then solve the x-direction equation for the normal force and substitute into the y-equation, finding:

\begin{large}\[ ma \cot\theta \cos\theta - mg = -ma\sin\theta\]\end{large}

Combining the acceleration terms by finding a common denominator and then using the trig identity

\begin{large}\[ \sin^{2}\theta + \cos^{2}\theta = 1\]\end{large}

gives the result:

\begin{large}\[ g = \frac{a}{\sin\theta} \] \end{large}

Method 2 – Tilted Coordinates

We have already drawn the relevant diagrams above, so we can simply write the equations:

\begin{large}\[ \sum F_{x} = mg\sin\theta = ma_{x} = ma \]
\[ \sum F_{y} = N - mg\cos\theta = ma_{y} = 0 \]\end{large}

The x-direction equation then immediately gives:

\begin{large} \[g\sin\theta = a\]\end{large}