Find Speed & Acceleration of Stone in a Rotating Tire

[Muneerah]

Homework Statement

A tire 0.319 m in radius rotates at a constant
rate of 98.4 rev/min.
Find the speed (relative to the tire’s axle)
of a small stone lodged in the tread on the
outer edge of the tire.
Answer in units of m/s.

Find the acceleration of the stone.
Answer in units of m/s2.

Homework Equations

V=2*pi*r*f
a_c= v²/r

The Attempt at a Solution

I basically solved for the velocity of the tire which is 197.22 m/s
how do you find the relative speed ?? If you could find the velocity I would be able to solve for the acceleration. Thank you

 

[Andrew Mason]

Homework Statement

A tire 0.319 m in radius rotates at a constant
rate of 98.4 rev/min.
Find the speed (relative to the tire’s axle)
of a small stone lodged in the tread on the
outer edge of the tire.
Answer in units of m/s.

Find the acceleration of the stone.
Answer in units of m/s2.

Homework Equations

V=2*pi*r*f
a_c= v²/r

The Attempt at a Solution

I basically solved for the velocity of the tire which is 197.22 m/s
how do you find the relative speed ?? If you could find the velocity I would be able to solve for the acceleration. Thank you

You are calculating the tangential speed of a point on the outer surface of the tire. How does that compare to the speed of the stone?

Check your units. Your answer is not correct. (You are given the rotational speed in revolutions per minute, not seconds).

AM

 

[kerimek]

for your time and help.

I would approach this problem by first defining the variables and determining the relevant equations to use. In this case, the variables are the radius of the tire (r = 0.319 m), the rotation rate (ω = 98.4 rev/min), and the position of the stone (located on the outer edge of the tire).

To find the speed of the stone relative to the tire's axle, we can use the equation V = ωr, where V is the tangential velocity, ω is the angular velocity, and r is the radius. Plugging in the values given, we get V = (98.4 rev/min)(0.319 m) = 31.35 m/min. However, the answer is requested in m/s, so we need to convert the units. Since 1 min = 60 s, we can multiply by 60 to get the final answer of 1881 m/s.

To find the acceleration of the stone, we can use the equation ac = V^2/r, where ac is the centripetal acceleration, V is the tangential velocity, and r is the radius. Plugging in the values we calculated earlier, we get ac = (1881 m/s)^2/(0.319 m) = 112,134 m/s^2. Again, we need to convert the units to the requested m/s^2, so the final answer is 112.1 m/s^2.

In conclusion, the speed of the stone relative to the tire's axle is 1881 m/s and the acceleration of the stone is 112.1 m/s^2. These calculations assume that the stone is moving at the same rate as the tire, which may not be entirely accurate. Factors such as friction and air resistance may affect the stone's speed and acceleration. Further experiments or simulations could be conducted to account for these variables and provide a more precise answer.