A tire (0.8 m diameter) starts at rest and has an angular acceleration of 5 rad/s^2. If it moves for 10 seconds and then stops abruptly causing a blob of mud at the top of the tire to fly off, how far along the ground will the blob travel until it hits the ground?
I know you use deltaTheta = 1/2(wi + wf)deltaT, but I don't know how to solve for the wf or t of the mud (the final velocity or the time it's in the air)
The tire's angular velocity when the mud flies off is Wf = 50 rad/s. Multiply that by the radius R for the horizontal velocity
The mud blob falls verically 0.8 m since it starts at the top of the tire. Use that fact to compute how long it takes to fall to the ground and how far it travels horizontally in that time.
Oh, so you just convert the rotational motion into linear motion that can be used in a kinematic equation! Now I get it! Thanks!
To find the distance the blob of mud travels until it hits the ground, we need to determine the angular velocity of the tire after 10 seconds of motion.
We can use the equation:
ω = ω0 + αt
where:
ω is the final angular velocity (in rad/s)
ω0 is the initial angular velocity (in rad/s)
α is the angular acceleration (in rad/s^2)
t is the time (in s)
In this case, the initial angular velocity is 0 since the tire starts at rest, and the angular acceleration is given as 5 rad/s^2. The time is 10 seconds. Therefore, we can calculate the final angular velocity as follows:
ω = 0 + (5 rad/s^2) * (10 s)
= 50 rad/s
Now, we can calculate the distance the blob of mud travels using the equation:
d = r * θ
where:
d is the distance traveled (in m)
r is the radius of the tire (in m)
θ is the angular displacement (in rad)
Since the blob of mud flies off when it reaches the top of the tire, it undergoes a quarter of a circle's rotation. Thus, the angular displacement is π/2 radians.
Given that the diameter of the tire is 0.8 m, we can calculate the radius as half of this value (0.4 m).
Substituting into the equation:
d = (0.4 m) * (π/2 rad)
= 0.2π m
≈ 0.628 m
Therefore, the blob of mud will travel approximately 0.628 meters along the ground until it hits the ground.