Math Problem Statement
Consider a car that has come to a complete stop at a red light. When the light turns green, the car begins to accelerate forward with a constant acceleration a=1.4 m/s2.
The tires on the car, which have radius r=0.33 m, roll without slipping while the car accelerates.
What will be the angular velocity of the car's wheels after they have completed at total of five rotations?
(a)
9.38 rad/s
(b)
6.51 rad/s
(c)
16.33 rad/s
Solution
To determine the angular velocity of the car's wheels after five rotations, we can follow these steps:
-
Convert Rotations to Distance: First, calculate the linear distance the car has traveled after five rotations of the wheels.
where .
So, for five rotations:
-
Use Kinematics to Find Linear Velocity: Given the car starts from rest and accelerates with a constant acceleration , we can use the following kinematic equation:
where:
- is the final linear velocity,
- (initial velocity),
- ,
- (distance traveled).
Plugging in the values:
-
Relate Linear Velocity to Angular Velocity: Since the wheels roll without slipping, we can relate the linear velocity of the car to the angular velocity of the wheels using:
Solving for :
So, the correct answer is:
(c) 16.33 rad/s.
Would you like a further breakdown, or do you have any questions?
Here are some related questions to expand on this concept:
- How would the solution change if the acceleration was doubled?
- What would be the impact if the car started with an initial velocity instead of from rest?
- How would frictional forces affect the rolling motion if it were no longer rolling without slipping?
- What is the relationship between linear and angular acceleration in rolling motion?
- How would the result differ if the car traveled over a distance of 15 meters instead?
Tip: When dealing with rolling motion, always check if there's slipping. Rolling without slipping creates a direct relationship between linear and angular quantities.
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Math Problem Analysis
Mathematical Concepts
Kinematics
Rotational Motion
Linear and Angular Velocity
Formulas
Circumference of a circle: C = 2πr
Linear distance: s = number of rotations × circumference
Kinematic equation for velocity: v² = u² + 2as
Linear to angular velocity conversion: v = rω
Theorems
-
Suitable Grade Level
Grades 11-12
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