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
(a) To calculate the angular velocity, we need to convert the rotation rate from revolutions per minute to radians per second.
1 rpm = (2 * π) rad/min.
Therefore, 2400 rpm = 2400 * (2 * π) rad/min = 4800 * π rad/min.
Now we need to convert this to rad/s:
1 min = 60 s.
So, 4800 * π rad/min = (4800 * π) / 60 rad/s = 80 * π rad/s.
The angular velocity is 80π rad/s.
(b) The linear speed (v) of a point on the edge of the grinding wheel can be calculated using the formula:
v = r * ω,
where r is the radius of the grinding wheel, and ω is the angular velocity.
The radius of the grinding wheel is half its diameter, which is 0.62 m / 2 = 0.31 m.
So, the linear speed is:
v = 0.31 m * 80π rad/s ≈ 77.92 m/s.
The radial (centripetal) acceleration (a) can be calculated using the formula:
a = r * ω^2.
So, the radial acceleration is:
a = 0.31 m * (80π rad/s)^2 ≈ 193867.6 m/s².
Thus, the linear speed is approximately 77.92 m/s, and the radial acceleration is approximately 193867.6 m/s².
To answer these questions, we need to understand the relationship between angular velocity and linear speed.
The angular velocity (ω) is the rate at which the wheel rotates, measured in radians per second (rad/s). The linear speed (v) is the speed of a point on the edge of the wheel, measured in meters per second (m/s). These two quantities are related by the formula:
v = ω * r
where r is the radius of the grinding wheel.
Now let's solve the questions:
(a) Calculate the angular velocity in rad/s:
The formula to convert rpm (revolutions per minute) to rad/s is:
ω = (2π * n) / 60
Given that the grinding wheel rotates at 2400 rpm, we can substitute that into the formula:
ω = (2π * 2400) / 60
Calculating this gives us:
ω = 100π rad/s
So the angular velocity is 100π rad/s.
(b) Calculate the linear speed and radial acceleration:
The radius of the grinding wheel is half of its diameter, so the radius would be 0.62 / 2 = 0.31 m.
Using the formula v = ω * r, we can substitute the values we have:
v = (100π rad/s) * (0.31 m)
Calculating this gives us:
v ≈ 97.4 m/s
So the linear speed of a point on the edge of the grinding wheel is approximately 97.4 m/s.
The radial acceleration is the acceleration of a point on the edge of the grinding wheel towards the center. It can be calculated using the formula:
ar = ω^2 * r
Substituting the values we have:
ar = (100π rad/s)^2 * (0.31 m)
Calculating this gives us:
ar ≈ 9612 m/s^2
So the radial acceleration of a point on the edge of the grinding wheel is approximately 9612 m/s^2.
To calculate the angular velocity of the grinding wheel in rad/s, you can use the formula:
Angular velocity (in rad/s) = (2π * N) / 60
where N is the rotational speed in rpm.
(a) Let's plug in the values:
Angular velocity = (2π * 2400) / 60 = 400π rad/s
So, the angular velocity of the grinding wheel is 400π rad/s.
To find the linear speed of a point on the edge of the grinding wheel, you can use the formula:
Linear speed (in m/s) = (angular velocity) * (radius of the grinding wheel)
(b) The radius of the grinding wheel is half its diameter, which is 0.62 m / 2 = 0.31 m.
Plugging in the values:
Linear speed = (400π rad/s) * (0.31 m) = 124π m/s
So, the linear speed of a point on the edge of the grinding wheel is 124π m/s.
To find the radial acceleration of a point on the edge of the grinding wheel, you can use the formula:
Radial acceleration (in m/s^2) = (linear speed)^2 / (radius of the grinding wheel)
Plugging in the values:
Radial acceleration = (124π m/s)^2 / (0.31 m) = (15376π^2) m/s^2
So, the radial acceleration of a point on the edge of the grinding wheel is (15376π^2) m/s^2.