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A wheel rotates with a constant acceleration of 2.0 rad/s2 . If the wheel starts from rest, the number of revolutions it makes in the first ten seconds will be approximately:
- a)8
- b)32
- c)24
- d)16
Correct answer is option 'D'. Can you explain this answer?
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A wheel rotates with a constant acceleration of 2.0 rad/s2 . If the wh...
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Understanding the Problem
The wheel starts from rest, meaning its initial angular velocity (ω₀) is 0 rad/s. It is undergoing constant angular acceleration (α) of 2.0 rad/s². We need to calculate the total number of revolutions made in 10 seconds.
Key Formulas
To find the angular displacement (θ) in radians, we can use the equation of motion:
- θ = ω₀t + 0.5αt²
Where:
- θ = angular displacement in radians
- ω₀ = initial angular velocity (0 rad/s)
- α = angular acceleration (2.0 rad/s²)
- t = time (10 seconds)
Calculation Steps
1. Substituting the Values:
θ = 0 * 10 + 0.5 * 2 * (10)²
2. Simplifying:
θ = 0 + 0.5 * 2 * 100
θ = 1 * 100
θ = 100 radians
3. Converting Radians to Revolutions:
To convert radians to revolutions, use the conversion factor (1 revolution = 2π radians):
Number of revolutions = θ / (2π)
Number of revolutions = 100 / (2 * 3.14) ≈ 15.92
Rounding to the nearest whole number gives approximately 16 revolutions.
Conclusion
Thus, the wheel makes approximately 16 revolutions in the first ten seconds, confirming that the correct answer is option 'D'.
The wheel starts from rest, meaning its initial angular velocity (ω₀) is 0 rad/s. It is undergoing constant angular acceleration (α) of 2.0 rad/s². We need to calculate the total number of revolutions made in 10 seconds.
Key Formulas
To find the angular displacement (θ) in radians, we can use the equation of motion:
- θ = ω₀t + 0.5αt²
Where:
- θ = angular displacement in radians
- ω₀ = initial angular velocity (0 rad/s)
- α = angular acceleration (2.0 rad/s²)
- t = time (10 seconds)
Calculation Steps
1. Substituting the Values:
θ = 0 * 10 + 0.5 * 2 * (10)²
2. Simplifying:
θ = 0 + 0.5 * 2 * 100
θ = 1 * 100
θ = 100 radians
3. Converting Radians to Revolutions:
To convert radians to revolutions, use the conversion factor (1 revolution = 2π radians):
Number of revolutions = θ / (2π)
Number of revolutions = 100 / (2 * 3.14) ≈ 15.92
Rounding to the nearest whole number gives approximately 16 revolutions.
Conclusion
Thus, the wheel makes approximately 16 revolutions in the first ten seconds, confirming that the correct answer is option 'D'.
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