Thinking Mathematically: Explore the quantitative dependencies of the acceleration upon the speed and the radius of curvature. Then answer the following questions. a. For the same speed, the acceleration of the object varies _______ (directly, inversely) with the radius of curvature. b. For the same radius of curvature, the acceleration of the object varies _ (directly, inversely) with the speed of the object. c. As the speed of an object is doubled, the acceleration is (one-fourth, one-half, two times, four times) the original value. d. As the speed of an object is tripled, the acceleration is (one-third, one-ninth, three times, nine times) the original value. e. As the radius of the circle is doubled, the acceleration is (one-fourth, one-half, two times, four times) the original value. f. As the radius of the circle is tripled, the acceleration is ______ (one-third, one-ninth, three times, nine times) the original value.

In circular motion there must be an acceleration towards the center of the circle, it is called cenripetal acceleration, in this case all the energy supplied to the system is used to change the direction of the speed even when its magnitude remains constant.

[tex]a_c = \frac{v^2}{r}[/tex]

Where [tex]a_c[/tex] the centripetal acceleration, v is the speed and r the radius of curvature of the circle.

Now we can answer the questions about centripetal acceleration.

A) For the same speed, the acceleration varies INVERSELY with the radius of curvature.

B) For the same radius of curvature the acceleration varies DIRECTLY with the speed.

C) The speed is doubled

v = 2 v₀

[tex]a_c = \frac{(2v_o)^2 }{r}[/tex]

[tex]a_c = 4 \ \frac{v_o^2 }{r}[/tex]

The acceleration becomes four times greater than the original value

D) The speed is tripled

v = 3 v₀

[tex]a_c = 9 \frac{ v_o^2}{r}[/tex]

Acceleration becomes nine times greater than the original

E) the radius of curvature is doubled

r = 2 r₀

[tex]a_c = \frac{v_o^2}{2 r_o }[/tex]

[tex]a_c = \frac{1}{2} a_o[/tex]

Acceleration is reduced to half the original value

F) The radius of curvature is tripled

r = 3 r₀

[tex]a_c = \frac{v_o^2 }{3 r_o} \\ \\a_c = \frac{1}{3} a_o[/tex]

The acceleration is reduced to a third of the initial one.

In conclusion using the expression for the centripetal acceleration we can find the answers for the questions are:

A) The acceleration varies INVERSELY with the radius of curvature.

B) The acceleration varies DIRECTLY with the speed.

C) The acceleration becomes four times greater.

D) The acceleration becomes nine times greater.

E) The acceleration is reduced to half.

F) The acceleration is reduced to a third.

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