Math Problem Statement

A roller coaster car moves in a vertical circle of radius R . At the top of the circle the car has speed v1 , and at the bottom of the circle it has speed v2 , where v2>v1 . Suppose that v2 is in the +x -direction, and +y -axis is directed upward. In terms of v1 and v2 , what is the ratio arad,bottom/arad,top ? Express your answer in terms of v1 and v2 .

Solution

We are tasked with finding the ratio of the radial accelerations at the bottom and top of a vertical circular motion, given the speeds v1v_1 at the top and v2v_2 at the bottom of the circle.

Key points:

  • Radial acceleration is given by the equation: arad=v2Ra_{\text{rad}} = \frac{v^2}{R} where vv is the speed of the object and RR is the radius of the circular path.

Radial acceleration at the top of the circle:

  • At the top of the circle, the car's speed is v1v_1, so the radial acceleration is: arad,top=v12Ra_{\text{rad,top}} = \frac{v_1^2}{R}

Radial acceleration at the bottom of the circle:

  • At the bottom of the circle, the car's speed is v2v_2, so the radial acceleration is: arad,bottom=v22Ra_{\text{rad,bottom}} = \frac{v_2^2}{R}

Ratio of radial accelerations:

Now, we find the ratio of the radial acceleration at the bottom to that at the top: arad,bottomarad,top=v22Rv12R=v22v12\frac{a_{\text{rad,bottom}}}{a_{\text{rad,top}}} = \frac{\frac{v_2^2}{R}}{\frac{v_1^2}{R}} = \frac{v_2^2}{v_1^2}

Thus, the ratio of the radial accelerations is: v22v12\boxed{\frac{v_2^2}{v_1^2}}

Summary:

The ratio of the radial acceleration at the bottom of the circle to the top is v22v12\frac{v_2^2}{v_1^2}.


Would you like more details or clarification on any part of this solution?

Here are five related questions to explore this concept further:

  1. How would the tension in the string (or normal force on the car) change between the top and bottom of the circle?
  2. How does gravity affect the car’s speed at different points in the circle?
  3. What would happen to the car's speed if friction were considered?
  4. How can energy conservation help us relate v1v_1 and v2v_2?
  5. How would the centripetal force differ at the top and bottom?

Tip: In circular motion, radial acceleration is always directed towards the center, and its magnitude depends on both the speed and the radius of the path.

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Math Problem Analysis

Mathematical Concepts

Circular Motion
Kinematics
Radial Acceleration

Formulas

arad = v^2 / R
arad, bottom / arad, top = v2^2 / v1^2

Theorems

Centripetal Force Theorem
Newton's Second Law (applied to circular motion)

Suitable Grade Level

Grades 10-12 (Physics)