A grinding wheel 0.62 m in diameter rotates at 2400 rpm.
(a) Calculate its angular velocity in rad/s.
rad/s
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel?
linear speed m/s
radial acceleration m/s2
To solve this problem, we need to convert the given values into appropriate units and use appropriate formulas. Here's how you can find the answers to the given questions:
(a) Calculate the angular velocity in rad/s:
Angular velocity is the rate at which an object rotates. It is usually measured in radians per second (rad/s). To calculate angular velocity, we can use the formula:
Angular Velocity = (2π * rotational speed) / 60
In this case, the rotational speed is given as 2400 rpm (revolutions per minute). To convert it to rotations per second, we divide by 60:
Rotations per second = 2400 rpm / 60 = 40 rotations/s
Now, we can calculate the angular velocity:
Angular Velocity = (2π * 40) rad/s = 80π rad/s ≈ 251.33 rad/s
So, the angular velocity of the grinding wheel is approximately 251.33 rad/s.
(b) Calculate the linear speed and acceleration of a point on the edge of the grinding wheel:
The linear speed of a point on the edge of the grinding wheel can be calculated using the formula:
Linear Speed = Angular Velocity * Radius
In this case, the diameter of the grinding wheel is given as 0.62 m. To calculate the radius, we divide the diameter by 2:
Radius = 0.62 m / 2 = 0.31 m
Now, we can calculate the linear speed:
Linear Speed = 251.33 rad/s * 0.31 m = 77.9853 m/s ≈ 78.0 m/s
So, the linear speed of a point on the edge of the grinding wheel is approximately 78.0 m/s.
The radial acceleration of a point on the edge of the grinding wheel can be calculated using the formula:
Radial Acceleration = (Linear Speed)^2 / Radius
Using the calculated values from earlier:
Radial Acceleration = (78.0 m/s)^2 / 0.31 m ≈ 19782.26 m/s^2 ≈ 1.98 * 10^4 m/s^2
So, the radial acceleration of a point on the edge of the grinding wheel is approximately 1.98 * 10^4 m/s^2.