newtons law question

Question

a 15kg box is sitting at the top of a 10m ramp angled at a 28degrees. assuming there is no friction, what will be the acceleration down the ramp?

Answer 1

Please note that the goal in the following is not to give away the answer, but to invite interested readers to engage with the question from a slightly different (and hopefully instructive) perspective.

When first answering a question of this sort, it can sometimes be helpful to begin with a brief review of the relevant concepts, to come up with a mental model of how you expect the system to behave, and then to make a cursory outline of an approach that might work.

We already know that Newton's laws are relevant (as stated in the title), and it might be worth briefly reviewing the meaning of these in this context. To some extent, we also must be comfortable with the idea of a 'ramp', and of the ramp being frictionless.

Intuitively, if you have ever gone sledding or skiing, you might consider the feeling immediately after unbreaking, just as you begin to accelerate down the slope. As the slope becomes increasingly shallow, the acceleration becomes less and less perceptible (a similar phenomenon has also been observed in incorrectly parked cars.) Because of this, there probably exists a relationship between the acceleration and the slope of the ramp: the steeper the angle, the faster the acceleration.

Newton's Laws allow us to determine (and/or predict) this relationship in a quantitative way. You may have encountered the notion of a 'free body diagram', and this conceptual framework can be especially useful in solving problems of this sort. The idea is to enumerate (and maybe draw or graph) all forces that act on the system, before adding them together in order to determine the net force. Once the net force is known, the acceleration can be found by applying Newton's 2nd Law.

To apply Newton's laws using a free body diagram or otherwise, we'll need a working definition of frictionless ramps. To do this, for lack of appropriate lab equipment we shall apply the at-times-treacherous (c.f. Aristotelian mechanics) but nonetheless tried and true method of thought experiment. The first step is to determine what we mean by a ramp: here, a ramp will refer to a smooth, planar surface or linear track, possibly tilted at an angle with respect to the direction of gravity. For example, a straight slide or flat hillside would count as 'ramps'. Next, if we haven't been told already, we might need to find out what it means for the ramp to be frictionless. Essentially, the force exerted by a frictionless ramp on an object resting on that ramp is always perpendicular to the surface of the ramp, or along its normal vector.

Now that we have an idea for how to approach the problem, and have familiarized ourselves with the background concepts, let's start our first attempt at an answer.

First, we can enumerate all the forces that act on the 15kg box: (bold letters indicate vectors)

1. The normal force (exerted by the frictionless ramp), called 'F_N'
2. The force of gravity (directed downwards), called mg
3. Other forces as yet unaccounted for (assumed to be non-existent in theory, but which could appear in experiments [especially sloppy ones.])

(It may help to draw a picture at this stage.) Next, we have the freedom to choose a coordinate system, and this freedom can be used if we are so inclined ultimately to make our approach more elegant and graceful. Since there are only two forces involved here, we may choose to align our coordinates with either force direction. For now, let us align the 'y' direction with the normal vector, so that the coordinate system is tilted at the same angle as the ramp.

In this coordinate system, the components of the normal force are simple: 0 (Newtons) in the x direction, and F_N = length(F_N) in the y direction.

To determine the x and y components of the gravitational force, we will need a few tools from math (trig and geometry in particular.) We know that the slope is at an angle of 28 degrees. I assume this is measured from the (gravitational) horizontal, or from the (counter-gravitational) vertical/normal force if measuring the direction of the normal vector, but please correct me if I am mistaken.

At this stage, readers might be asking themselves the following:

1. What does the 28 degree slope of the ramp imply about the direction of gravity in the given coordinate system?
2. How can SOHCAHTOA be used to determine the components of the gravitational force?
3. (Optional) What must the acceleration be in the 'y' direction (i.e. normal to the ramp?) What does this tell us about the normal force?
4. What force is it that causes acceleration along the 'x' direction (i.e. along the surface of the ramp?)
5. What coordinate system should the answer be expressed in? With respect to the ramp, or with respect to the direction of a plumb line (i.e. the 'normal' normal vector opposite gravity), or with respect to either? (If with respect to the normal normal vector, or lab frame, how can geometry and trigonometry be used to determine the vector along the ramp's surface?)