The roller coaster in Figure P6.36 starts with a velocity of 15.4 m/s. One of the riders is a small girl of mass 30 kg. Find her apparent weight when the roller coaster is at locations B and C. At these two locations, the track is circular with the radii of curvature given in the figure. The heights at points A, B, and C are hA= 24.85 m, hB = 34.2 m, and hC=0 m.
To find the apparent weight of the small girl at locations B and C, we need to consider the forces acting on her at each location. The apparent weight is the force experienced by an object in a non-inertial frame of reference, such as when riding a roller coaster.
At location B, the girl is at a height of hB = 34.2 m and the roller coaster track has a radius of curvature indicated in the figure. We can start by finding the acceleration of the girl at this location using the centripetal acceleration formula:
aB = v^2 / rB
where v is the velocity of the roller coaster and rB is the radius of curvature at location B. From the information provided, v = 15.4 m/s.
Now, let's find the radius of curvature at location B using the given information:
rB = hB + rA
where rA is the radius of curvature at location A. We need to subtract hA from hB to find the difference in heights and then use the fact that the change in gravitational potential energy is equal to the change in kinetic energy:
m * g * (hB - hA) = 1/2 * m * v^2
where m is the mass of the girl, g is the acceleration due to gravity, and v is the velocity of the roller coaster. From the information provided, m = 30 kg.
Now, we can solve for rA:
rA = (2 * m * g * (hB - hA) / v^2) - rB
Substitute the given values into the equation to find rA.
With the radius of curvature at location B (rB) and the acceleration at location B (aB), we can find the net force acting on the girl:
Fnet = m * aB
With the net force, we can find the apparent weight of the girl at location B:
Apparent weight at B = m * g - Fnet
Now, let's move on to location C. At location C, the girl is at a height of hC = 0 m, which means she is at the bottom of the roller coaster loop. The radius of the curvature at this location is given in the figure.
We can find the acceleration at location C using the same centripetal acceleration formula:
aC = v^2 / rC
where v is the velocity of the roller coaster and rC is the radius of curvature at location C.
With the acceleration at location C, we can find the net force acting on the girl:
Fnet = m * aC
Finally, we can find the apparent weight of the girl at location C:
Apparent weight at C = m * g - Fnet
By following these steps, you can find the apparent weight of the small girl at locations B and C on the roller coaster.