Which will win the race?

Classical Mechanics Level 2

As the image shows, $O$ is the center of a circle, $E$ is at the bottom of the circle, and $AE,BE,CE,DE$ are all smooth inclines along the circle.

Now, let's put four balls at rest at points $A,B,C,D.$

If whichever gets to point $E$ first wins the race, ignoring all the frictions, which ball will win the race?

The ball at A The ball at B The ball at C The ball at D They will get to E at the same time It depends on other conditions

1 solution

Laszlo Mihaly
Jan 18, 2018

The distance between D and E will be called $s$ and the distance between B and E is $2R$ . The angle between the two lines is $\alpha$ . The DBE triangle is a right triangle, therefore $s=2R \cos \alpha$ .

The time it takes to reach the bottom is $t=\sqrt{2s/a}$ , where the acceleration is $a=g \cos\alpha$ . Inserting $s$ and $a$ we get

$t=\sqrt{\frac{2s}{a}}=\sqrt{\frac{4R \cos \alpha}{g\cos \alpha}}= \sqrt{\frac{4R}{g}}$ , independent of the angle $\alpha$ . Therefore all of the balls will get to the bottom at the same time.

Why is this question level 1? I thought it would be assess as level 3 question.

Kelvin Hong - 3 years, 4 months ago

I have just looked at it and it is now upgraded to 3.

Laszlo Mihaly - 3 years, 4 months ago

Which will win the race?

1 solution

0 pending reports