A remote-controlled car’s wheel accelerates at
22.7 rad/s.
If the wheel begins with an angular speed of 11.0 rad/s, what is the wheel’s angular speed after exactly twenty full turns?
To find the wheel's angular speed after exactly twenty full turns, you need to know the angular acceleration of the wheel. However, the angular acceleration is not provided in the given information. Hence, it is currently not possible to calculate the wheel's angular speed after twenty full turns.
If the angular acceleration were given, you could use the kinematic equation for rotational motion:
ω = ω₀ + αt
where:
- ω is the final angular speed
- ω₀ is the initial angular speed
- α is the angular acceleration
- t is the time
Since the angular acceleration is not given, it is not possible to provide a specific answer to the question at this time.
To find the wheel's angular speed after twenty full turns, we need to consider the initial angular speed and the acceleration.
Given:
Initial angular speed (ω₁) = 11.0 rad/s
Angular acceleration (α) = 22.7 rad/s
We can use the formula to relate the final angular speed (ω₂) with the initial angular speed and the angular acceleration:
ω₂ = ω₁ + αt
Here, t represents the time. Since we want to calculate the angular speed after twenty full turns, we need to determine the time it takes to complete twenty full turns.
One full turn in radians is equal to 2π radians. So, twenty full turns would be equal to:
20 turns * 2π radians/turn = 40π radians
To calculate the time, we can use the formula:
θ = ω₁t + 0.5αt²
Substituting the given values:
40π radians = 11.0 rad/s * t + 0.5 * 22.7 rad/s * t²
Simplifying the equation:
40π = 11.0t + 11.35t²
Now, we need to solve this quadratic equation. Bringing all terms to one side, we get:
11.35t² + 11.0t - 40π = 0
Using the quadratic formula, t can be determined as:
t = (-b ± √(b² - 4ac)) / (2a)
Here, a = 11.35, b = 11.0, and c = -40π.
Plugging in the values:
t = (-11.0 ± √((11.0)² - 4 * 11.35 * (-40π))) / (2 * 11.35)
Calculating this equation will give us two possible values for t. However, we are only interested in the positive value because the time cannot be negative.
By calculating this equation, the positive value of t is approximately 2.553 seconds.
Now, plug in this value into the angular speed formula to calculate the final angular speed:
ω₂ = ω₁ + αt
ω₂ = 11.0 rad/s + 22.7 rad/s * 2.553 seconds
By evaluating this expression, the final angular speed after exactly twenty full turns is approximately 68.212 rad/s.