A 34.1 kg child rides a Ferris wheel whose diameter is 18.3 m. When the ride is turning at a constant rate, the child’s speed has a constant value of 4.23 m/s. Find the normal force the seat exerts on the child at:
a) The top of the ride
b) The bottom of the ride
here are answers, don't know what steps were taken to get this though, like formulas, just need a walk through
a. 267N
b. 401N
a= v²/R.
top
ma= mg –N
N=m(g-a) =
=m (g- v²/R) =
= 34.1{9.8 - (4.23²/9.15 )}=
=267 N.
bottom
ma=N-mg
N= m(g+a) =
= m (g+ v²/R) =
= 34.1{9.8 +(4.23²/9.15 )}=
=401 N
To find the normal force at the top and bottom of the ride, we need to consider the forces acting on the child.
Let's start with the forces at the top of the ride:
a) Top of the ride:
At the top of the ride, the child is moving in a circular path. There are two forces acting on the child: the weight force (mg) downward and the normal force (N) upward.
The weight force is given by:
Weight force = mass × gravity
Weight force = 34.1 kg × 9.8 m/s^2
Weight force = 333.98 N
The centripetal force acting on the child is provided by the normal force. At the top of the ride, the centripetal force is the net inward force and equals:
Centripetal force = mass × velocity^2 / radius
Centripetal force = 34.1 kg × (4.23 m/s)^2 / (18.3 m / 2)
Centripetal force = 34.1 kg × 17.8929 m^2/s^2 / 9.15 m
Centripetal force = 66.79 N
Since the centripetal force is provided by the normal force, we have:
Normal force = Centripetal force
Normal force = 66.79 N
Therefore, the normal force at the top of the ride is 66.79 N.
b) Bottom of the ride:
At the bottom of the ride, the forces acting on the child are the weight force (mg) and the normal force (N). However, this time the weight force acts upward and the normal force acts downward to provide the centripetal force.
The weight force remains the same:
Weight force = 333.98 N
The centripetal force is still given by:
Centripetal force = mass × velocity^2 / radius
Centripetal force = 34.1 kg × (4.23 m/s)^2 / (18.3 m / 2)
Centripetal force = 66.79 N
Since the centripetal force is now provided by the downward normal force, we have:
Normal force = Centripetal force + Weight force
Normal force = 66.79 N + 333.98 N
Normal force = 400.77 N
Therefore, the normal force at the bottom of the ride is 400.77 N.
So, the normal force at the top of the ride is 66.79 N and at the bottom, it is 400.77 N.