Patrick Teaches Physics

What are "cos" and "sin"?  They're buttons on a calculator -- right?  You enter an angle, push the button, and some number appears.  Somehow -- if you remember your trig rules (SOH, CAH, TOA) -- you can use this number get the side lengths of triangles. I think the above pretty much sums up the typical student's understanding, and it's adequate to complete most of the tasks a physics student has to with trig functions.  But it doesn't have to be so mysterious, and it can be very helpful if it isn't.  So in this post I'll explain what these functions are and how they are related. Consider trying to specify a point on the unit circle: We could use the Cartesian coordinates x and y, or we could use the angle θ. Both have their merits, and in practice we might want to go back and forth between them.  So, given an angle θ, what are x and y?  We call the functions that answer that question sine and cosine (usually written as sin and cos for sho

Physics has a reputation for being a difficult subject.  I think that's partly because of the focus on problem solving -- which is an inherently high-level activity -- and partly because of the pride of place it gives to mathematics -- which makes it like studying two subjects at the same time. But much of what a physics student is asked to do (at least in high school and first year uni) is not difficult.  Instead it's the relatively simple business of learning vocabulary, and showing you know what the word means. Consider the following question: A ball (mass 120g) travelling in a straight line at 18 m/s bounces off a wall, returning at the same speed. Calculate the change in momentum of the ball. This question will neatly divide students into two groups: those who can define "momentum" and those who can't*.  For the first group (who know that the momentum of an object is the product of its mass and velocity) this will be easy marks (the answer is -4.3 kgm/s, ass

You grind your way through a problem, get an answer, and want to know if it is correct.  How can you tell? In the previous post I talked about changing the angle in a problem (the inclined plane) to see how that affected other angles.  It can help you see how the angles relate to each other.  In this post I will use the same technique, but this time to test if my answer could be wrong. Let's return to the inclined plane.  Suppose a block of mass m = 2.5 kg slides down a frictionless plane at a slope of θ = 35 degrees to the horizontal.  You need to determine the block's acceleration.    You calculate an acceleration of a = 8.0 m/s/s down the slope.  Your friend, attempting the same problem, gets an answer of a = 5.6 m/s/s.  Who is right? After both checking that your calculators are set to degrees (not radians), you look at each other's work.  You have a = g cos θ while your friend has a = g sin θ where g is the usual acceleration due to gravity.  At least one of

How often have I walked into a lecture theatre planning to talk about some deep secret of physics only to end up arguing with students about which angle is which in a problem?  To be fair, I also often struggled to follow geometrical arguments when I was a student.  One trick I learned was to try to see the problem as dynamic rather than static, by which I mean imagine the angles changing so I can see how they relate to each other. For example, consider the inclined plane.  In these problems, some object is placed on a surface that is at an angle to horizontal: something sliding down a ramp, for example. We have the usual free-body diagram, indicating the relevant forces (weight of the object, normal force, friction), and the angle of the slope is labelled θ (theta).  To determine the acceleration (if any) of the object we need the net force acting on it.  And one of the things that will require is the angle between the normal force (N) and the vertical.  It turns out to be the same as

When I first encountered the radian it seemed that the world had just become unnecessarily more complicated.  What was wrong with plain old degrees , and how did I know when to use which?  I found out it really helps to learn what a radian actually is . Consider a circle of radius r , and imagine wanting to know the length of an arc, s , subtended by an angle θ. Suppose the angle is such that s = r.  That angle actually defines the unit radian, so in that case θ = 1 radian.  Obviously, the arc length is proportional to the angle, so we get the general formula  s = rθ so long as we measure the angle in radians.  That's what makes the radian so useful: it gives a direct relationship between angular distance and length.  One place that has obvious applications is in rotational kinematics, because it allows us to easily relate angular velocity to tangential velocity. But how big is a radian in degrees?  The circumference of a circle is the arc length for the whole circle (360 degre

When solving physics problems we always have the benefit of choosing our frame of reference.  Not only can we chose where the origin of our coordinate is, but we can also chose which part of our system is stationary.  Consider the following simple kinematic problem: A car is travelling along, and we want to know how long it takes to travel some distance.  We can plug numbers into a kinematic equation and get the answer.  If the car is accelerating then the problem is mathematically a bit more fiddly (a quadratic equation, or do it in two steps with two equations) but it's not conceptually harder. But now consider a similar problem with two moving bodies: Now there's two cars at different speeds, and the question is: how long will it take the red car to overtake the blue car?  This looks rather harder because the distance the red car has to travel is not immediately clear.  But that's only because of our choice of reference frame, which up till now has been implicit. We inst