Math Problem Statement
Problem 1: A speck of dust moves along a circular path at constant speed. The radius of the circle is 1.01m , and the distance traveled by the speck of dust as it moves from point A to point B is 3.79m .
Part (a) What is the angular displacement of the speck of dust, in radians, as it travels from A to B ? (The diagram is for visualization purposes only. The calculated angle might not appear to match the apparent angle in the diagram.) Numeric : A numeric value is expected and not an expression. θ = __________________________________________rad
Part (b) If it takes the speck of dust 0.76s to travel from A to B , what is the magnitude of its angular speed in units of radians per second? Numeric : A numeric value is expected and not an expression. ω = __________________________________________rad/s
Part (c) What is the magnitude of the average angular acceleration of the speck of dust in units of radians per second squared? Numeric : A numeric value is expected and not an expression. α = __________________________________________rad/s2
Solution
To solve this problem, we’ll tackle each part separately.
Part (a): Angular Displacement ()
The angular displacement in radians can be found using the formula: where:
- is the arc length (distance traveled along the circular path) from point A to point B, which is ,
- is the radius of the circle, given as .
Substituting the values:
Calculating:
So, the answer for Part (a) is:
Part (b): Angular Speed ()
The angular speed is defined as the rate of change of angular displacement over time. Since the dust particle moves at a constant speed, we can calculate using: where:
- is the angular displacement we just calculated, ,
- is the time taken, given as .
Substituting the values:
Calculating:
So, the answer for Part (b) is:
Part (c): Average Angular Acceleration ()
Since the dust particle moves at a constant speed along the circular path, there is no change in angular speed, meaning there is no angular acceleration. Thus:
So, the answer for Part (c) is:
Summary of Answers:
- (a) Angular Displacement:
- (b) Angular Speed:
- (c) Angular Acceleration:
Do you have any questions or would you like a deeper explanation for any part?
Related Questions to Expand Understanding:
- How does changing the radius affect the angular displacement if the distance traveled remains the same?
- What would be the angular displacement if the radius were halved?
- If the speed of the particle varied over time, how would you calculate the angular acceleration?
- How is linear speed related to angular speed in circular motion?
- How would you calculate the centripetal acceleration of the dust particle?
Tip:
In circular motion, angular displacement in radians is directly proportional to the arc length, simplifying calculations when moving between points on a circular path.
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Math Problem Analysis
Mathematical Concepts
Circular motion
Angular displacement
Angular speed
Angular acceleration
Formulas
Angular displacement formula: θ = s / r
Angular speed formula: ω = θ / t
Angular acceleration formula: α = Δω / Δt
Theorems
Relationship between linear distance and angular displacement
Suitable Grade Level
Grades 11-12
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