Phy 201
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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We previously had this exchange:
I am having some trouble with a few of the test questions for my upcoming tests and was hoping you could clarify a few things:
1)A Ferris wheel with radius 24 meters is moving fast enough that at the top of its arc a 150-lb person at the rim of the wheel has an apparent weight of only 63 pounds.
• How fast does a point on the rim of the wheel move?
• What will be the apparent weight of the same person at the bottom of the arc?
• What will be the apparent weight when the person is at the same height as the center of the wheel?
For this questions I understand that observed weight will be less at the top and more on the bottom due to centripital forces combination with normal force. I do not understand how to solve it however. Since observed weight is given in lbs, should i just convert this to Newtons and then solve for solve for velocity using F=v^2/r(m). If so would I use 150lbs converted to kg for mass and what would i use for force?
Then how would i get observed weight on the bottom?
The person's apparent weight is the force exerted on the person by the seat.
The net force is the sum of the force of the seat and the force of gravity (i.e., the person's weight).
The net force is equal to the centripetal force -m v^2 / r.
So we could write
F_seat + F_gravity = -m v^2 / r, so that
-v^2 / r = F_seat / m + F_gravity / m.
You could of course convert the forces to metric units and solve for v, and you should probably do so.
A more elegant solution is to note that F_gravity / m = -m g / m = -g, the negative of the acceleration of gravity (we implicitly have assumed that the upward direction is positive, which we declare by noting that we have done so).
The magnitude of F_seat is 63/150 the magnitude of the weight, and is directed upward. Thus
F_seat / m = 63/150 * m g / m = 63/150 g.
Thus
-v^2 / r = 63/150 g - g
so that
v^2 = r * (87/150) * 9.8 m/s^2 and
v = sqrt( 24 m * 87/150 * 9.8 m/s^2) = 12 m/s, approx.
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I still dont understand parts of this problem. Lbs and kgs aren't forces so how am i supposed to solve the equation F=m*v^2/r for v if i dont have force. Do i just take the observed weight*9.8m/s/s to get Force and use the observed weight in kg for mass? If this is the case how am I supposed to find observed weight for the bottom. I am having a lot of trouble with this question.
The pound is a measure of force, with 1 pound = 9.8 / 2.2 Newtons.
The kg is a measure of mass.
You don't need to remember this, but in the British system the measure of mass is the slug. 1 slug * 1 ft / s^2 = 1 pound. Since the acceleration of gravity is 32 ft / s^2, the weight of a 1-slug mass at the surface of the Earth is 1 slug * 32 ft/s^2 = 32 pounds.
Now to the problem. Let's go back to the following:
F_seat + F_gravity = -m v^2 / r, so that
-v^2 / r = F_seat / m + F_gravity / m.
You could of course convert the forces to metric units and solve for v, and you should probably do so.
You need to understand two things:
F_seat + F_gravity is the net force on the person, and
when moving at the top of a circle of radius r at velocity v the net force on the person is -m v^2 / r.
these two expressions for the net force on the person give you the equation
F_seat + F_gravity = -m v^2 / r.
You don't really need to understand any more than this. You can easily convert the forces to metric units if you wish, so we know everything in this equation but v. We can therefore can solve for v. We get
v = sqrt( - r / m * (F_seat + F_gravity) )
Making the conversions:
150 pounds * (9.8 N / (2.2 lb)) ) = 660 Newtons, approx., and similarly we find that 63 lbs = 270 Newtons, again approximately.
660 N is the weight of the person, so 660 N = m g and m = 660 N / g = 660 N / (9.8 m/s^2) = 67 kg, approx..
F_seat is an upward force and F_gravity is a downward force, so we get
v = sqrt( - 24 m / (67 kg) * (270 N - 660 N) ) = ... etc.
It wasn't necessary to make the conversion, and the original note discusses this, but the main idea is that the net force is the centripetal force, and the net force is the sum of the upward seat force and the downward acceleration of gravity. At the top of the circle the centripetal force, being toward the center, is downard.
At the bottom of the circle the only difference is that the centripetal force, again being toward the center, is now upward. So we have
F_seat + F_gravity = +m v^2 / r.
This time, the unknown quantity is F_seat. F_gravity is still downward, and so according to our choice of positive direction is negative. Thus
F_seat = m v^2 / r - F_gravity = m * v^2 / r - (-660 N) = m v^2 / r + 660 N.
Substituting m, r and the v we calculated previously we find F_seat, which if converted back to pounds should turn out to be 237 lb.