Riding a Loop-the-Loop: A car in an amusement park ride rolls without friction a

Question

Riding a Loop-the-Loop: A car in an amusement park ride rolls without friction around the track shown in the figure below. 7-32fig.gif It starts from rest at point A at a height h above the bottom of the loop. Treat the car as a particle.

(a) What is the minimum value of h (in terms of R) such that the car moves around the loop without falling off at the top (point B)?____ R

(b) If h = 4.10R and R = 27.0 m, compute the speed, radial acceleration, and tangential acceleration of the passengers when the car is at point C, which is at the end of a horizontal diameter. ______ m/s (speed) ______ m/s2 (radial acceleration -- toward center of circle) ______ m/s2 (tangential acceleration -- downward)

Explanation / Answer

Ptential Energy of Car at Point A = mgh
P.E = mgh

At Point B , Car has Potential Energy + Kinetic Energy

K.E = 0.5 * mv^2 + m*g*2*R
Now,
Minimum speed needed at B -
mv^2/R - m*g = 0
v = sqrt(g*R)


For a Minimum value of h.
mgh = 0.5 * mv^2 + m*g*2*R
g*h = 0.5 * g*R + g*2*R
h = 5/2*R

Minimum Value of h = 5/2 * R

(b)
R = 27.0 m
h = 4.10 *R

Using Energy Conservation -
K.Ea + P.Ea = K.Ec + P.Ec
0 + m*g*h = K.Ec + m*g*R
K.Ec = m*g*(4.1 - 1)R
0.5*v^2 = g * 3.1 * R
v = sqrt(2*9.8*3.1*27)
v = 40.5 m/s

Radial acceleration, aR  = v^2/R
aR = 40.5^2 / 27 m/s^2
aR =  60.75 m/s^2
Radial acceleration , aR = 60.75 m/s^2

Only Downward Force is Gravitational Force, Therefore
Tangential Accleration , aT = 9.8 m/s^2

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