Combined Variation Word Problem, please help my head is spinning

Question

The force needed to keep a car from skidding on a curve varies inversely as the radius of the curve and jointly as the weight of the car and the square of the car's speed. Suppose that 245 pounds of force keeps a 2000-pound car from skidding on a curve of radius 600ft at 35mph. What force would keep the same car from skidding on a curve of radius 900ft at 45mph?

Answer 1

This is a classic joint variation problem.

Step 1: Assign variables to your unknowns: F=force, w=weight of car, s=speed and r=radius.

Step 2. Your "k" or constant of variation is ALWAYS in the numerator.

Step 3: Solve for "k" first.

Step 4: Plug "k" back into your original equation and solve for your unknown, "F" (note: this is not the same "F" as the first half of the problem since some variables are changing).

So, if k and s vary "jointly", this means they vary "directly". Both variables go on the opposite side of the "F". Usually, you start by writing F=...

Also, your radius varies "inversely". This means you put "r" in the denominator under "k" and "s"

F=(k*w*s²)/r Set up-"k" in

245=k*(2000)*35² Plug in what you know from the first part of the problem

600

147000=k*(2000)*35² Divide both sides by 2000 and 35² to isolate the "k"

k=0.06 Constant of variation

Last, you are being asked to find the force with r=900 and s=45. Now that we have "k" and "w" remains the same since we are dealing with the same car, thus:

F=(0.06)*(2000)*45² Set up and solve for F. Cross multiply

900

900*F=0.06*2000*45² Multiply all terms on the right side

900*F=243000 Divide by 900

F=270 ANSWER

Answer 2

To understand this variation problem, it helps to know the full equation for centripedal force, F. The equation is explicitly F = mv²/r. We can arrive at this result, since we know that force varies jointly by the weight (or almost equivalently, mass, m) of the car and the car's speed squared, or v², and also inversely by the radius, r.

Now we need to consider what is changing and what is not. The car's 2000 lb weight is not changing, as they don't indicate it. Speed and radius are changing, but by what factor? In this cases, a factor is a number that you multiply onto an initial value to get the final value. Equivalently it is final divided by initial. Here we can show that the speed squared increases by a factor 45²/35²= 1.653. But radius also increases, by a factor of 900/600=1.5. Therefor, by what factor does F change? Since v² is in the numerator of our main equation, we multiply our initial F by 1.653. Since r is in the denominator, we also divide F by 1.5. Does this result in an increase or decrease in force? Since 1.653/1.5 =1.102 is greater than 1, the force will increase. We can see that if we do (245 lb) x 1.102 which gets the answer.

Notice we did not need to use the weight of the car at all in our calculation. That's not because it doesn't matter in physics, but rather because it does not change here (that is to say its factor is =1). This is the power of dealing with factors of change.