A ferris wheel, 24.0m in diameter rotates once every 12.5s. What is the person’s apparent weight (mass = 80kg) at the top and at the bottom, compared to the person’s weight at rest?

Answer 1

An object with no net force acting on it will travel in a straight line at constant velocity. (If the object is at rest, then the constant velocity is equal to zero.)

An object traveling in a circle is not traveling in a straight line. Therefore, there must be a net force acting on it that keeps it moving in a circle. This force is called a "centripetal force," and is given by the equation

Fc=mv^2/r

The problem told us that

r=24.0 m

f=1/12.5 revolutions/second

m=80 kg

The frequency of the revolutions and the radius can be used to find the velocity using

v=2*pi*f*r = 12.06 m/s

making the centripetal force on the person

Fc=80 * 12.06^2/24 = 485.1 N

The centripetal force is the net force. A net force is the sum of all the forces acting on the object. A person sitting in a Ferris wheel has two forces acting on it that sum together to make this net force. One force is gravity. Gravity is negative because it always points down towards the ground, and is always equal to

-Fg=-m*g

where g=9.81 m/s2 is a constant for the earth. That makes

-Fg=-784.8 N

The other force acting on the person is called the "normal force," FN The normal force is the contact force between the person and the ferris wheel. The normal force is used to find the "apparent weight," mA

FN=mA*g

mA=FN/g

So the "apparent weight" tells you how hard the floor is pushing back on you. If the person is standing still, then its net force must be zero, which means the normal force must be equal to the force of gravity.

FNet=FN-Fg=0

FN=Fg

Gravity pulls the person down towards the center of the earth, but the surface pushes back with the same force as gravity so that the person doesn't move. When we solve this equation, we find that the apparent weight is equal to the actual weight.

FN=Fg

mAg=mg

mA=m

But when the person is moving and the net force is not equal to zero, the apparent weight will be different than the actual weight.

For the person on the Ferris wheel, the net force is equal to the centripetal force. The centripetal force always points towards the center of the Ferris wheel. When the person is at the bottom of the wheel, the center of the Ferris wheel is up, so the centripetal force is positive.

FNet=+Fc=FN-Fg

FN=+Fc+Fg= 485.1 N + 784.8 N = 1269.9 N

mA=FN/g=129.5 kg

The motion of the Ferris wheel makes your apparent weight feel larger than your actual weight when you are at the bottom of the wheel.

When the person is at the top of the wheel, the center of the Ferris wheel is down, so the centripetal force is negative.

FNet=-Fc=FN-Fg

FN=-Fc+Fg= -485.1 N + 784.8 N = 299.7 N

mA=FN/g=30.5 kg

The motion of the Ferris wheel makes your apparent weight feel smaller than the actual weight when you are at the top of the wheel.

You have probably felt this effect when you drive fast down a hilly road. When you are at the top of a hill, it often feels like you are "flying" or like you are almost weightless. This is because your apparent weight is smaller than your actual weight on the top of a hill, just like when you are at the top of a Ferris wheel.