A Level Maths: Forces In Two Dimensions
Introduction
What we are going to look at in this article is a few examples that involve forces along a rough horizontal plane as well as look at how we can solve problems where the plane is an inclined slope.
In order to understand the questions it is going to be taken that you understand how to resolve forces parallel and perpendicular to the plane whether an object is travelling horizontally or whether an object is moving on an inclined plane.
As part of A Level Maths Mechanics you must understand how forces work and how to apply them when writing down the equations of motion.
We will look at three examples that show you the various different stages and the working that you need to be able to show clearly. As always, you need to make sure that you produce a clear force diagram.
Example
To answer this question the first thing that you need to do is to draw a force diagram. It does not need to be a work of art, but it needs to be clear and it needs to show all the appropriate forces that are acting on the object.
There are two parts to the question. The first part requires you to determine the acceleration of the object. To do this you need to apply F = ma and the standard convention is to take the right as the positive direction.
\begin{aligned} F=m a & \\ 45 \cos 50-8 & =8 a \\ a & =\frac{45 \cos 50-8}{8} \\ & =2.62 \mathrm{~ms}^{-2} \end{aligned}Now that we have the acceleration we can now determine the distance that has been travelled. This can be done using an appropriate SUVAT equation.
\begin{aligned} & u=0 \quad t=5 \quad a=2.62 \quad s=s \\ & s=u t+\frac{1}{2} a t^2 \\ & =\frac{1}{2}(2.62)(5)^2=32.75 \mathrm{~m} \end{aligned}Example
In this question we now have an object that is on an inclined slope. Again a force diagram is needed.
In the diagram above, the dashed shape object is the object after it has travelled a distance of 20m down the slope as per the question. You will see from the force diagram all relevant forces have been added to the diagram and since the object is sliding down the slope, downwards will be taken to be the positive direction.
Applying F = ma parallel to the plane we have:
\begin{aligned} F=m a & \\ 50 g \cos 65 & =50 a \\ a & =\frac{50 g \cos 65}{50} \\ & =4.14 \mathrm{~ms}^{-2} \end{aligned}The question wants us to determine the velocity after it has travelled 20m down the slope. Since the object was released from rest we know that the initial velocity is equal to zero. We can now apply SUVAT to help us determine the velocity:
\begin{aligned} & u=0 \quad v=v \quad a=4.14 \quad s=20 \\ & v^2=u^2+2 a s \\ & =2(4.14)(20) \\ & \therefore v=12.9 \mathrm{~ms}^{-1} \end{aligned}
Example
For this question the force diagram is shown below:
For part (i) of the question you need to take into account a resistance force of 30N. It is the force parallel to the plane that is causing the girl to slide down the slope. Applying F = ma parallel to the plane down the slope gives:
\begin{aligned} F=m a & \\ 35 g \cos 70-30 & =55 a \\ a & =\frac{55 g \cos 70-30}{55} \\ & =2.81 \mathrm{~ms}^{-2} \end{aligned}For part (ii) in order to find the speed of the girl of 5 seconds, it is a case of applying the correct SUVAT equation:
\begin{aligned} u=0 \quad v & =v \quad t=5 \quad a=2.8 \\ v & =u+a t \\ v & =2.8(5)=14.03 \mathrm{~ms}^{-1} \end{aligned}We have covered three examples here. These questions cover the basic principles that you need to be able to understand and to apply these principles to more complicated and challenging questions.
In future A Level Mechanics articles we will look at how Friction is calculated and how this is also incorporated into questions involving Newton’s Laws.
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