A car travels at a constant speed around a circular track whose radius is 2.8 km. The car goes once around the track in 430 s. What is the magnitude of the centripetal acceleration of the car?
acceleration=w^2*r=
w=2PI*r/430
To find the magnitude of the centripetal acceleration of the car, we can use the formula for centripetal acceleration:
\[ a = \frac{v^2}{r} \]
Where:
- \(a\) is the centripetal acceleration
- \(v\) is the velocity of the car
- \(r\) is the radius of the circular track
In this case, we are given the radius of the track (2.8 km) and the time it takes for the car to go once around the track (430 s). We need to find the velocity of the car in order to calculate the centripetal acceleration.
The velocity of the car can be found by dividing the distance traveled (the circumference of the circular track) by the time taken:
\[ v = \frac{2 \pi r}{t} \]
Where:
- \(v\) is the velocity of the car
- \(r\) is the radius of the circular track
- \(t\) is the time taken to go around the track
Plugging in the values, we get:
\[ v = \frac{2 \pi (2.8 \text{ km})}{430 \text{ s}} \]
Now, we can substitute the value of \(v\) into the centripetal acceleration formula:
\[ a = \frac{\left(\frac{2 \pi (2.8 \text{ km})}{430 \text{ s}}\right)^2}{2.8 \text{ km}} \]
Simplifying the equation, we get:
\[ a \approx \frac{11.2 \pi^2 \text{ km}^2}{430 \text{ km} \cdot \text{s}^2} \]
Finally, we can evaluate the expression to find the magnitude of the centripetal acceleration.