CP Riding a Loop-the-Loop. A car in an amusement park ride rolls without friction around a track (Fig. P7.42). The car 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)? (b) If ?h? = 3.50 R and ?R? = 14.0 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. Show these acceleration components in a diagram, approximately to scale.
Solution 46P Introduction At top the car will not fall if the velocity at the top of the hoop is such that the centrifugal force is equal to the gravity. We will use this information to calculate the velocity of the car at the top point. From the velocity we can calculate the energy at that point. The required energy will be supplied by the potential energy because of the starting height of the car. We will calculate the required height from the required potential energy of the car. Now by using the given values we will calculate the velocity and acceleration at the specified point. Step 1 The car will not fall from the top of the hoop, if the velocity of the car at the top of the hoop is such that, the centrifugal force is equal to the weight of the car. Suppose v is the velocity of the car at the top point. Hence the centrifugal force is Here m is the mass of the car. Now the weight of the car is Hence we have Hence the kinetic energy at that point is And the potential energy of the car at that point is Hence the total energy at that point is Now if the starting height of the car is h, then we have Hence the height h will be 2.5R.