Patrick Teaches Physics

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

When we start learning physics in school, I think we're usually told to convert all quantities into SI base units (metres, kilograms and so forth).  This isn't bad advice as it can certainly prevent some problems.  But it's not always optimal.  And at some point, when we delve more deeply into a particular branch of physics, we will need to become comfortable with somewhat "niche" units. A simple example of sticking to non-base units is the measurement of density.  Suppose I am given a small cuboid of some material and asked to measure its density.  I get a ruler and measure its dimensions: 3.5 cm x 2.2 cm x 4.7 cm I put the sample on an electric balance and find its mass to be 120 g These units -- centimetres and grams -- are what my tools use.  If I convert into metres and kilograms, here's what my calculation looks like: If instead I'm a bit more sophisticated, I can stay in my original units: The second method is more efficient: there's simply less

There is probably no habit that a physics student can learn that has a higher return on investment than including units in calculations.  To see what I mean, consider this routine momentum calculation: Can't remember the units for momentum?  Don't have to, they're already there on the page!  This is particularly useful if you are not using base SI units: you don't get confused or forget which units you're using along the way. Better than that, this method gives us a powerful way to check our own calculations.  In the next example, I calculate the time required for an object to fall a distance: What!?  Metres per second isn't a unit of time!  I've made a silly mistake re-arranging the equation; the kind of mistake it's easy for a student to make, and even pros do occasionally.  But pros tend to catch such mistakes because they have kept track of the units.  That means we can fix it: The unit being right gives us some confidence the number is right.