Question

A car in an amusement park ride rolls without friction around a track (Figure 1). The car starts from rest at point A at a height h above the bottom of the loop. Treat the car as a particle.

Figure 1:

PART 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)?

PART B:

If the car starts at height h= 4.30 Rand the radius is R1 = 16.0 m ,compute the speed of the passengers when the car is at point C, which is at the end of a horizontal diameter.

PART C:

Compute the radial acceleration of the passengers when the car is at point C, which is at the end of a horizontal diameter.

PART D:

Compute the tangential acceleration of the passengers when the car is at point C, which is at the end of a horizontal diameter.

Answer 1

Let the mass of the car is m.� The minimum speed (v) of this car at the point b is given by mv^2/R = mg = > v^2 = Rg� So, the total energy of the cart at B = mg(2R) + 1/2 mv^2 = 2mgR + 1/2 mgR = 5/2 mgR� So, from conservation of energy� mgh = 5/2 mgR = > h = 5/2 R �h = 4.3 R and R = 16 m = > h = 4.3 times 16 = 68.8 m� Let the speed at 'c' is v_1.� Then from conservation of energy� 1/2 mv^2_1 + mgR = mgh = 4.3 mgR� on v^2_1 = 6.6gR = > v_1 = [6.6 times 9.8 times 16]^1/2 = 32.17 m/s� therefore v_1 = 32.17 m/s� The radial acceleration at C = v^2_1/R = 6.68R/R = 6.6g = 64.68 m/s^2� Tangential acceleration at C = gravitational acceleration = g� = 9.8 m/s^2