Calculating Radius for Stone in Rotating Tire with Static Friction

[TastyTyr]

please help...uniform circular motion..I am STUMPED

A stone has a mass of 3.0 10-3 kg and is wedged into the tread of an automobile tire, as the drawing shows. The coefficient of static friction between the stone and each side of the tread channel is 0.71. When the tire surface is rotating at 16 m/s, the stone flies out of the tread. The magnitude FN of the normal force that each side of the tread channel exerts on the stone is 1.8 N. Assume that only static friction supplies the centripetal force, and determine the radius r of the tire.

 

[Doc Al]

Show what you've done so far. (Hint: What's the maximum possible centripetal force that the friction can supply?)

 

[quicknote]

I would first start by identifying the relevant equations and principles that can be applied to this problem. In this case, we can use the equations for circular motion and the concept of centripetal force. The equation for centripetal force is Fc = mv^2/r, where Fc is the centripetal force, m is the mass of the object, v is the velocity, and r is the radius of the circular motion.

In this problem, we are given the mass of the stone (3.0 10-3 kg), the velocity of the tire (16 m/s), and the normal force (1.8 N). We can also assume that the only force acting on the stone is the force of static friction, since the stone is not sliding out of the tread channel.

To solve for the radius r, we can rearrange the equation for centripetal force to solve for r: r = mv^2/Fc. We can then substitute in the given values and solve for r:

r = (3.0 10-3 kg)(16 m/s)^2/(1.8 N)

r = 0.533 m

Therefore, the radius of the tire is approximately 0.533 meters. This means that the stone was located at a distance of 0.533 meters from the center of the tire when it flew out.

It's important to note that this calculation assumes that the stone was rotating at the same angular velocity as the tire. If the stone was rotating at a different angular velocity, the calculation would be more complex and would require additional information. Additionally, this calculation only takes into account the forces acting on the stone in the horizontal direction. Other forces, such as gravity, may also be acting on the stone and could affect the final result. Further analysis and experimentation would be needed to fully understand the dynamics of the stone in this rotating tire system.