Answer:
46.9 m/s
Explanation:
When the cart is at the bottom of the loop, there are two forces acting on the person:
- The force of gravity, acting downward, of magnitude [tex]mg[/tex] (where m is the mass and g is the acceleration due to gravity)
- The normal reaction exerted on the person, acting upward, of magnitude N
Their resultant must be equal to the centripetal force, so the equation of the forces at the bottom of the loop is:
[tex]N+mg=m\frac{v^2}{r}[/tex]
where v is the speed of the cart at the bottom of the loop and r the radius of the loop.
N represents the apparent weight of the person in the cart: here we are told that the person must feel 3 times as heavy, this means that
[tex]N=3mg[/tex]
Substituting into the equation,
[tex]3mg+mg=m\frac{v^2}{r}[/tex]
Which means
[tex]v=\sqrt{4gr}[/tex]
Here the radius of the loop is
r = 56.0 m
So, the speed needed is:
[tex]v=\sqrt{4(9.8)(56.0)}=46.9 m/s[/tex]
