Question

2 part question- help please!
a. What is the maximum speed (in m/s) a vehicle (mass = 555 kg) can have at the top of a bump on a roller coaster (radius = 16.3 m) in order that it remain on the track?
b.A pail of water is rotated in a vertical circle of radius 1.26 meters. What must be the minimum speed (in m/s) of the pail at the top of the circle in order for no water to spill out?
Thanks so much!!!

Answer 1

b)
The forces on the bucket at the top of the loop (if it's going *just fast enough to keep the water in the bucket are
\[\sum F_y = mg=ma_{centripetal}\]
\[g=a_{centripetal}\]
\[g=\frac{v^2}{r}\]
With that you can solve for the velocity

Answer 2

For part a, you need to set up the forces at the top of the track as

\[ \sum F = N-mg=-ma_{cent}\]
and the normal force has to be greater than zero for the car to still be on the track
\[0<N=-ma+mg\]
\[0<m(-a+g)\]
\[g>a\]
\[\frac{v^2}{r}<g\]

The velocity has to be less than some value, so you solve for the max. velocity.


B)
Looking at our original expression for velocity,
\[g=\frac{v^2}{r}\]
In order for the water not to fall out of the bucket, its centripetal acceleration has to be *greater than or equal to* the gravity
\[g≤\frac{v^2}{r}\]

The velocity has to be greater than some value, so you solve for the minimum velocity.

Answer 3

THANKS SO MUCH! this helped me so much! you rock.

Answer 4

^_^

Answer 5

Does it all make sense?

Answer 6

yes very much so. thanks again!