# Uniform Circular Motion of dice

• uchicago2012
In summary: You're doing good, but I think you're overthinking it a bit. Just remember that the net force in the x-direction is mv2/R, and you'll get there eventually.In summary, the problem involves a car rounding a curve with a radius of 275 meters and fuzzy dice hanging at an angle of 12° from the vertical. The homework equations for this problem include a = v2/R for uniform circular motion. The attempt at a solution involves considering the forces acting on the dice and setting up equations for the net force in the x and y directions. The net force in the x-direction is equal to the tension force, which can be found using trigonometry and the centripetal force equation. Ultimately, the
uchicago2012

## Homework Statement

Fuzzy dice hang from the rear-view mirror of a car rounding a curve. If the curve has a radius of 275 meters and the dice are hanging at an angle of 12° from the vertical, how fast is the car going?

## Homework Equations

a = v2/R
for an acceleration with a constant magnitude, as in uniform circular motion. I took this to mean that a in this equation is the direction of the acceleration. i.e., the Cartesian coordinates.

## The Attempt at a Solution

In uniform circular motion, the magnitude of the acceleration is always the same and pointing inwards. Does this mean the magnitude of the acceleration is equal to R? So the magnitude of acceleration would be 275 m/s2 in this case? I'm not sure how to begin this problem but I could solve it if that was true. But I think R and the magnitude of the acceleration are just related, not necessarily equal, but I don't really understand how.

uchicago2012 said:

## Homework Equations

a = v2/R
for an acceleration with a constant magnitude, as in uniform circular motion. I took this to mean that a in this equation is the direction of the acceleration. i.e., the Cartesian coordinates.

## The Attempt at a Solution

In uniform circular motion, the magnitude of the acceleration is always the same and pointing inwards. Does this mean the magnitude of the acceleration is equal to R? So the magnitude of acceleration would be 275 m/s2 in this case?
No, R is not 275 m/s2, it is 275 m. Plus as you already wrote a=v2/R, and that is not equal to R.
I'm not sure how to begin this problem but I could solve it if that was true. But I think R and the magnitude of the acceleration are just related, not necessarily equal, but I don't really understand how.
Try drawing a free-body diagram for the dice. What forces act on the dice?

Consider the forces acting on the dice when it is inclined at an angle with the vertical.
There is tension in the string, its weight vertically downwards and lastly the centripetal force.
Equate horizontal and vertical forces and solve the 2 equations.
You will get the the answer.

So I tried something but it didn't really work out:
F = centripetal force
T = tension force
Fg = force gravity = mg
Theta = 12 degrees, measured from the vertical

Fnet,x = F + T sin theta
Fnet,y = T cos theta - mg

so if I arrange the equations like so:
T sin theta = -F
T cos theta = mg

T sin theta = Tx and T cos theta = Ty and by trig, Tx/Ty = tan theta
so I said

tan theta = -F/mg
and since F = centripetal force, which is also F = m * (v2/R)
I then can say
tan theta = -m(v2/R) / mg
The negative sign becomes a problem, since I must take the square root of the term to solve for v. So should my F be pointing in the opposite direction of T, which would then make everything work out nicely? Or is my method just flawed? I rather feel like T and F should be pointing in the same direction (both towards the center of the circle), but I'm not sure what I did wrong.

You're pretty close.
uchicago2012 said:
So I tried something but it didn't really work out:
F = centripetal force
T = tension force
Fg = force gravity = mg
Theta = 12 degrees, measured from the vertical

Fnet,x = F + T sin theta
Fnet,y = T cos theta - mg
The y equation is okay, but we have to think about the x equation. It's best to think about what is physically causing the forces: there is only the tension force T and gravitational force mg.

The centripetal force F is the net force and is the vector sum of the two forces I just mentioned. Another way to say that is: Fnet is equal to mv2/R, acting in the x-direction:

mv2/R = Fnet,x = T sin(θ)
0 = Fnet,y = T cos(θ) - mg

(On the left side of each equation, I have written what we know must be the net force, due to the circular motion of the dice. On the right side are the forces that come from some physical cause--string tension or gravity).

so if I arrange the equations like so:
T sin theta = -F
That one should be slightly different; no minus sign.
T cos theta = mg
Yes.

## 1. What is uniform circular motion?

Uniform circular motion is a type of motion in which an object moves in a circular path at a constant speed. This means that the object covers equal distances in equal amounts of time, resulting in a constant velocity.

## 2. How does uniform circular motion apply to dice?

In the context of dice, uniform circular motion refers to the rolling of a die along a circular path with a constant speed. This can be observed when rolling a die on a flat surface, as it will roll in a circular motion until it comes to a stop.

## 3. What is the relationship between radius and velocity in uniform circular motion?

The relationship between radius and velocity in uniform circular motion is known as the "tangential velocity." This relationship states that the velocity of an object moving in a circular path is directly proportional to the radius of the circle and the angular velocity (how quickly the object rotates around the circle).

## 4. How does centripetal force play a role in uniform circular motion of dice?

Centripetal force is the force that keeps an object moving in a circular path. In the case of dice rolling, the centripetal force is provided by the friction between the die and the surface it is rolling on. This force constantly changes direction to keep the die moving in a circular path.

## 5. Can the motion of a rolling die ever be truly uniform?

In theory, yes, the motion of a rolling die could be truly uniform if there were no external forces acting on it. However, in reality, there will always be some amount of friction and other external forces that will affect the motion of the die, making it not truly uniform. This is known as ideal versus non-ideal uniform circular motion.

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