The drawing shows a chain-saw blade. The rotating sprocket tip at the end of the guide bar has a radius of 1.4×10-2 m. The linear speed of a chain link at point A is 3.0 m/s. Find the angular speed of the sprocket tip in rev/s.
To solve this problem, we need to relate the linear speed of the chain link to the angular speed of the sprocket tip.
First, let's define some key terms:
- Linear speed: The rate at which an object moves along a straight path.
- Angular speed: The rate at which an object rotates or moves in a circular path.
- Radius: The distance from the center of a circle to any point on its circumference.
Given information:
- Radius of the sprocket tip (r) = 1.4×10^-2 m
- Linear speed of the chain link at point A (v) = 3.0 m/s
The linear speed of the chain link is related to the angular speed of the sprocket tip by the formula:
v = rω
where:
- v is the linear speed
- r is the radius
- ω (omega) is the angular speed
To find the angular speed in revolutions per second (rev/s), we need to convert the linear speed and radius accordingly.
1. Convert the linear speed from m/s to m/rev:
To find the number of meters covered in one revolution, we need to find the circumference of the circle represented by the sprocket tip:
circumference = 2πr
In this case, the distance covered in one revolution is the same as the circumference:
distance/revolution = circumference = 2πr
Using the given radius (1.4×10^-2 m), we can calculate the distance covered in one revolution.
2. Apply the conversions and solve for angular speed:
Now, we can use the formula v = rω and the calculated distance/revolution value to find the angular speed.
angular speed (rev/s) = linear speed (m/rev) / radius (m)
Plugging in the calculated values, we can solve for the angular speed.