Problem 14-18

 

Problem Description

Analytical Solution

ADAMS Solution

ADAMS Model File

 

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Determine the height h to the top of the incline D to which the 200 kg roller coaster car will reach, if it is launched at B with a speed just sufficient for it to round the top of the loop at C without leaving the track. The radius of curvature at C is rC = 25m.

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Analytical Solution

Problem Description

ADAMS Solution

ADAMS Model File

 

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First, determine the velocity of the car at the top of the loop at C necessary to maintain contact with the tracks.

Drawing a free body diagram of the car at the top of the loop at C.

The equation of motion in the normal direction gives (neglecting the normal force)

(1) mg = mv2/r

Which gives vC = (rg)1/2 = 15.66 m/s

Now, we find the required initial velocity at B, using conservation of energy.

TB + VB = TC + VC

(2) 1/2 m vB2 + 0 = 1/2 m vC2 + mg hC

Substituting the values and solving the above equation gives

vB = 30.53 m/s

Now, to find the maximum height at D, we again apply conservation of energy.

TB + VB = TD + VD

(3) 1/2 m vB2 + 0 = m g hD

Solving equation (3) gives the maximum height as.

hD = 47.5 m     

 

ADAMS Solution

Problem Description

Analytical Solution

ADAMS Model File

 

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In the ADAMS simulation, the roller coaster is given the initial velocity of 30.53 m/s as calculated analytically.  The height, speed, and kinetic energy of the roller coaster are shown. Notice that 1. the velocity at the top of the loop at C is vC =  15.5 m/s. 2. The maximum height at D, corresponding to the point where the velocity is zero is h = 47.5m. 3. The kinetic energy is the opposite of the height. Since the only potential energy in the system is due to gravity, this reflects the principle of conservation of energy.

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