Roller coaster rider weight problem

[Cisneros778]

Homework Statement

Suppose the vertical loop has a radius of 6.81 m. What is the apparent weight (Wap) of a rider on the roller coaster at the bottom of the loop? (Assume that friction between roller coaster and rails can be neglected. Give your answer in terms of m and g.)

R = 6.81 m

Homework Equations

F=m(a_c)

The Attempt at a Solution

Since the roller coaster is at the bottom the centripetal acceleration should be smaller than the weight of the rider on the roller coaster... I think...

 

[PeterO]

Homework Statement

Suppose the vertical loop has a radius of 6.81 m. What is the apparent weight (Wap) of a rider on the roller coaster at the bottom of the loop? (Assume that friction between roller coaster and rails can be neglected. Give your answer in terms of m and g.)

R = 6.81 m

Homework Equations

F=m(a_c)

The Attempt at a Solution

Since the roller coaster is at the bottom the centripetal acceleration should be smaller than the weight of the rider on the roller coaster... I think...

For a start "centripetal acceleration" is an acceleration - units ms^-2 and "weight of the rider" is a force - units N so I am not sure what you mean that one will be less than the other. Like saying I a taller than you weigh!

Secondly - ever been in a car that drives through a dip? Did you feel lighter or heavier at the time?

 

[Cisneros778]

Sorry I meant centripetal force vs. the force of gravity. I don't really pay much attention while I'm in a car driving through loops. I reckon you are heavier.

 

[PeterO]

Sorry I meant centripetal force vs. the force of gravity. I don't really pay much attention while I'm in a car driving through loops. I reckon you are heavier.

That is correct, so from your own experience you already know that the answer to this question will be that your apparent weight is greater at the bottom.
The next difficulty is - how much bigger?

As far as the centripetal force is concerned, the size of that depends on how fast you are going. You do know that F_c = mv²/R

Now if you knew from what height the cart had come from you could use the transformation of PE to KE to work that out.

PE_top → KE_bottom

so mgh → 0.5mv²

so mv² = 2mgh → mv²/R = 2mgh/R