A particle is moving on a circular path of radius 0.5 m at a constant speed of 10 ms⁻¹. Find the time taken to complete 20 revolutions.
Understand the Problem
The question is asking us to find the time taken for a particle moving at a constant speed along a circular path to complete a certain number of revolutions. To solve this, we need to calculate the circumference of the circle, then use the speed to find the time for one revolution and multiply it by the total number of revolutions.
Answer
The total time taken for \( n \) revolutions is given by \( T_{total} = n \times \frac{2\pi r}{v} \).
Answer for screen readers
The total time taken can be written as:
$$ T_{total} = n \times \frac{2\pi r}{v} $$
Steps to Solve
- Calculate the Circumference of the Circle
The circumference of a circle can be calculated using the formula:
$$ C = 2\pi r $$
where ( r ) is the radius of the circle. If the radius is given, substitute its value into the formula to find the circumference.
- Find the Time for One Revolution
Next, we need to calculate how long it takes to complete one revolution. If the particle is moving at a constant speed ( v ), the time ( T ) for one revolution is given by:
$$ T = \frac{C}{v} $$
Substitute the value of the circumference ( C ) from step 1 and the given speed ( v ) into this equation.
- Calculate Total Time for Multiple Revolutions
To find the total time ( T_{total} ) taken for a specific number of revolutions ( n ), use the formula:
$$ T_{total} = n \times T $$
Substitute the value of ( T ) from step 2 into this equation and multiply by the total number of revolutions.
The total time taken can be written as:
$$ T_{total} = n \times \frac{2\pi r}{v} $$
More Information
This formula allows you to quickly find the time needed for a particle traveling in a circular path to complete multiple revolutions. Understanding the relationship between speed, circumference, and time is essential in physics and engineering.
Tips
- Forgetting to convert units if necessary (e.g., from cm to m).
- Not using the correct formula for circumference.
- Confusing the radius with the diameter of the circle.
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