# A pair of dice is hanging from the rear view window. The car makes a right turn. Find the radius of the circular path of the dice?

A pair of fuzzy dice is hanging from the rear view window on a massless string. While making a right turn at 40 mph, the fuzzy dice makes an angle of 23 degrees from the vertical. Find the radius of the circular path of the fuzzy dice.

I know that I can find the centripetal acceleration of the dice. But where do I go from there?

Please help.

### 2 Answers

- Steve4PhysicsLv 76 months ago
EDIT

See if this answers the question from your comment. Sorry it’s a bit long.

You take 40mph (=17.88 m/s) as the speed of the car and everything in it. The pair of dice are *stationary relative to the car*, so the dice are also moving at 40mph in a circular path.

To move in a circular path requires a force called the centripetal force (which equals mv²/r). For the dice (moving in a horizontal circle), the only horizontal force is the horizontal component of tension (Tsin(23º)). So Tsin(23º) is the centripetal force; that’s why we can write Tsin(23º) = mv²/r.

In fact, we are making an approximation. Because of the circular path, the speed is not 40mph everywhere inside the car. The speed is slightly less than 40mph on one side and slightly more than 40mph on the other side. But as long as car’s width is much smaller than the radius of the path, the effect is negligible. And the dice will be near the centre of the car anyway.

Hope that helps.

END EDIT

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If the mass of the dice is m, then the weight is mg. Vertically there is no acceleration, so the vertical component of tension and the weight balance:

Tcos(23º) = mg

T = mg/cos(23º)

If the speed is v (converted from mph to metres/second) then the centripetal force acting on the dice is the horizontal component of T (which is Tsin(23º)).

Tsin(23º) = mv²/r

Substituting for T:

(mg/cos(23º))sin(23º) = mv²/r

Cancelling m and using sin/cos = tan gives:

gtan(23º) = v²/r

r = v²/(gtan(23º))

- SparkyLv 66 months ago
wouldn't it be an equal and opposite reaction?

so the opposite of the 23 degrees to the left would be 23 degrees to the right equalling a radius of 23 degrees.

I don't know if that's correct, but that's how I would look at it.

But radius is in ft or inches or cm or meters, not degrees. If you knew the length of the string but it isn't given.

I've updated the main answer to try to explain.