A monster truck has tires with a diameter of 1.10 m and is traveling at 30.0 m/s. After the brakes are applied, the truck slows uniformly and is brought to rest after the tires rotate through 49.0 turns.
(a) What is the initial angular speed of the tires?
(b) What is the angular acceleration of the tires? (Enter the magnitude only.)
(c) What distance does the truck travel before coming to rest?
Given: D = 1.1m., V0 = 30m/s, 49rev.
C = pi*D = 3.14 * 1.1 = 3.46m. = Circumference of tirs.
a. Vo = 30m/s * 6.28rad/3.46m = 54.5rad/s.
b. d = 49rev * 3.46m/rev = 169.3m =
stopping dist.
a = (Vf^2 - Vo^2) / 2d,
a = (0 - (30)^2) / 338.6 = -2.66m/s^2 =
Linear acceleration.
a = -2.66m/s^2 * 6.28rad/3.46m =
-4.82rad/s^2. = Angular acceleration.
c. d = 49rev * 3.46m/rev = 169.3m.
(a) Well, to find the initial angular speed of the tires, we can use the formula:
Angular speed = (final angular displacement) / (time)
Since the truck comes to rest, the final angular displacement is 49.0 turns, which is equal to 49.0 * 2π rad.
Now, we need to find the time it takes for the truck to come to rest. The truck slows uniformly, so we can use the equation:
Final angular speed = Initial angular speed + (angular acceleration) * (time)
Since the final angular speed is 0 (truck comes to rest), we can rearrange the equation to solve for time:
time = (Final angular speed - Initial angular speed) / (angular acceleration)
Now, we have all the information to find the initial angular speed:
Initial angular speed = (final angular displacement) / (time)
= (49.0 * 2π rad) / ((Final angular speed - Initial angular speed) / (angular acceleration))
But since we know the final angular speed is 0, we can simplify the equation:
Initial angular speed = (49.0 * 2π rad) / (time / (angular acceleration))
Well, this equation is going in circles! So, we need to rearrange it again and isolate the initial angular speed:
Initial angular speed = (49.0 * 2π rad) * (angular acceleration) / time
Now we can plug in the given values and solve for the initial angular speed.
(b) To find the angular acceleration of the tires, we can substitute the values into the equation:
Final angular speed = Initial angular speed + (angular acceleration) * (time)
Since the final angular speed is 0, we are left with:
0 = Initial angular speed + (angular acceleration) * (time)
Now, we can rearrange the equation to solve for angular acceleration:
angular acceleration = -Initial angular speed / time
Again, we can plug in the given values and solve for angular acceleration.
(c) To find the distance the truck travels before coming to rest, we can use the formula:
Distance = ((Initial angular speed) * (time) + (1/2) * (angular acceleration) * (time)^2) * (radius)
Since the truck comes to rest, the initial angular speed is 0.
So, the equation simplifies to:
Distance = (1/2) * (angular acceleration) * (time)^2) * (radius)
Now, plug in the given values and solve for distance.
But hey, while this calculation might not be a walk in the park, at least you'll have a tire-iffic time with all those turns!
To solve this problem, we can use the equations of rotational motion.
(a) The initial angular speed of the tires can be found using the equation:
Angular speed (ω) = Final angular displacement (θ) / Time taken (t)
Since we know the final angular displacement (49.0 turns) and the time taken to stop (t), we can rearrange the equation to solve for the initial angular speed:
ω = θ / t
In this case, the final angular displacement is given as 49.0 turns, which is equivalent to 49.0 revolutions or 49.0 * 2π radians. The time taken (t) is the time required for the truck to come to a stop.
(b) The angular acceleration (α) of the tires can be found using the equation:
Angular acceleration (α) = Final angular speed (ωf) - Initial angular speed (ωi) / Time taken (t)
Since we know the final angular speed (0 rad/s) and the initial angular speed (ωi), we can rearrange the equation to solve for the angular acceleration:
α = ωf - ωi / t
(c) The distance the truck travels before coming to a rest can be found using the equation:
Distance (d) = Initial velocity (v) * Time taken (t) - 0.5 * Acceleration (a) * Time taken (t)^2
In this case, the initial velocity (v) is given as the speed of the truck (30.0 m/s) and the acceleration (a) can be calculated using the angular acceleration (α) and the radius of the tires (r).
Let's calculate:
(a) To find the initial angular speed:
ω = θ / t
ω = (49.0 * 2π) / t
(b) To find the angular acceleration:
α = ωf - ωi / t
Since the final angular speed is 0 rad/s, the equation simplifies to:
α = -ωi / t
(c) To find the distance:
First, we need to find the angular acceleration:
α = (ωf - ωi) / t
Next, we can calculate the acceleration:
a = α * r
Finally, we can find the distance traveled:
d = v * t - 0.5 * a * t^2
Let's calculate the answers using the given information.
To find the answers to these questions, we can use the equations of rotational motion. Let's break it down step by step:
(a) To find the initial angular speed of the tires, we can use the equation:
angular speed = (final angular position - initial angular position) / time
The final angular position is given as 49.0 turns, and we know that one full turn is equivalent to 2π radians. So, the final angular position in radians is:
final angular position = 49.0 turns * 2π radians/turn
The initial angular position is 0 radians, as that's where the tires start.
We are not given the time it takes to bring the truck to rest, so we have to calculate it. The time is given by:
time = distance / velocity
In this case, the distance traveled is the circumference of the tires, which is given by:
distance = 2π * radius
The radius of the tires is half of the diameter, so:
radius = 1.10 m / 2
Now, we can substitute the values into the equation for time:
time = (2π * radius) / velocity
Finally, we can put it all together to calculate the initial angular speed:
initial angular speed = (49.0 turns * 2π radians/turn - 0 radians) / ((2π * radius) / velocity)
(b) The angular acceleration is the rate at which the angular velocity changes with time. For uniformly slowing down rotation, the angular acceleration is constant.
We can find the angular acceleration using the equation:
angular acceleration = (final angular speed - initial angular speed) / time
Since we have already found the initial angular speed and calculated the time in part (a), we can substitute these values in to calculate angular acceleration.
(c) To find the distance the truck travels before coming to rest, we can use the equation:
distance = (angular speed^2 - initial angular speed^2) / (2 * angular acceleration)
We already have the values for angular speed, initial angular speed, and angular acceleration from parts (a) and (b). By substituting these values, we can calculate the distance.
I hope this explanation helps you to solve the problem!