Make it dynamic 2: Check the limits

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 you is wrong.  

This sort of thing is quite common: mis-identifying the relevant angle, or incorrectly applying the trig rules are both easy to do.  Which is where this sanity check is handy: recalculate the acceleration for an angle where you already know what the answer should be.

Suppose the angle were zero.  That's a flat horizontal surface, the acceleration should be zero.  Does your solution give that answer?  For θ = 0 your answer is

a = g

while your friend's is

 a = 0

You could also think about what happens when the angle is 90 degrees (surface is vertical).  Then the acceleration of the block should just be acceleration due to gravity.  Put θ = 90 into the two solutions and you will see that your solution gives the wrong answer, while your friend's is right.  Better luck next time!

These tests can't guarantee that a particular answer is right, but can help you spot a lot of wrong answers. 

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