A roller-coaster car speeds down a hill past point A where R1 = 10.4 m and then rolls up a hill past point B where R2 = 15.6 m, as shown below.

R1 and 2 are radii
(a) The car has a speed of 21.0 m/s at point A. if the track exerts a normal force on the car of 2.05 multiplied by 104 N at this point, what is the mass of the car?

(b) What is the maximum speed the car can have at point B for the gravitational force to hold it on the track?

To solve this problem, we will use the principles of circular motion and the law of conservation of energy.

(a) To find the mass of the car, we can use the centripetal force equation:

Fc = m * (v^2 / r)

where Fc is the centripetal force, m is the mass of the car, v is the velocity, and r is the radius of the circular path.

At point A, the centripetal force is provided by the normal force, which is perpendicular to the path of the car. Therefore, we have:

Fc = Normal Force = 2.05 * 10^4 N
v = 21.0 m/s
r = R1 = 10.4 m

Plugging these values into the equation, we get:

2.05 * 10^4 N = m * (21.0 m/s)^2 / 10.4 m

Now, let's solve for the mass:

m = (2.05 * 10^4 N * 10.4 m) / (21.0 m/s)^2

Calculating this expression will give us the mass of the car at point A.

(b) To find the maximum speed the car can have at point B, we need to consider the balance between the gravitational force and the centripetal force at that point.

At the highest point on the track (point B), the gravitational force and the centripetal force must be equal in magnitude. The gravitational force is given by:

Fg = m * g

where m is the mass of the car and g is the acceleration due to gravity.

The centripetal force at point B is provided by the normal force and is given by:

Fc = Normal Force

Equating these forces, we have:

m * g = Normal Force

To find the maximum speed, we need to calculate the value of g at that location and then solve for v in the equation:

m * g = Normal Force

where m is the mass calculated from part (a).

Once we find the value of g, we can calculate the maximum speed using the equation:

v = sqrt(r * g)

where r is the radius of the circular path at point B and g is the acceleration due to gravity.

Calculating these expressions will give us the maximum speed the car can have at point B for the gravitational force to hold it on the track.