A grinding wheel 0.3m in diametre has amass of 5kg is rotating at an angular velocity of 2300rev/min.what is the kinetic energy?
To find the kinetic energy of the rotating wheel, we need to use the formula for the rotational kinetic energy:
Kinetic Energy (KE) = (1/2) * moment of inertia * angular velocity^2
First, we need to calculate the moment of inertia (I) of the grinding wheel. The moment of inertia depends on the shape and mass distribution of the object.
For a solid disk rotating around its central axis, the moment of inertia can be calculated using the formula:
I = (1/2) * mass * radius^2
Since the grinding wheel is a solid disk, we can use this formula. Given that the diameter of the grinding wheel is 0.3m, we can calculate its radius (r) as follows:
radius = diameter / 2 = 0.3m / 2 = 0.15m
Next, we have the mass of the grinding wheel as 5kg.
Now we can substitute these values into the formula to find the moment of inertia (I):
I = (1/2) * mass * radius^2
= (1/2) * 5kg * (0.15m)^2
= (1/2) * 5kg * 0.0225m^2
= 0.05625 kg*m^2
Next, we need to convert the angular velocity from revolutions per minute (rev/min) to radians per second (rad/s).
To convert from rev/min to rad/s, we use the conversion factor 2π radians = 1 revolution, and 1 minute = 60 seconds.
angular velocity in radians/second (ω) = (angular velocity in revolutions/minute) * (2π radians / 1 revolution) * (1 minute / 60 seconds)
Now, we can calculate the angular velocity in radians per second (ω):
ω = 2300 rev/min * 2π radians / 1 revolution * 1 minute / 60 seconds
= 2300 * (2π/60) rad/s
= 241.9099 rad/s (approximately)
Finally, we substitute the values of the moment of inertia (I) and the angular velocity (ω) into the kinetic energy formula to find the kinetic energy (KE) of the grinding wheel:
KE = (1/2) * I * ω^2
= (1/2) * 0.05625 kg*m^2 * (241.9099 rad/s)^2
= 1651.87 Joules (approximately)
Therefore, the kinetic energy of the grinding wheel is approximately 1651.87 Joules.