A grinding wheel 0.28 m in diameter rotates at 2400 rpm. What is the acceleration of a point on the edge of the grinding wheel?
a=w^2 *r
in SI units, w= 2400rpm * 1min/60sec*2PIrad/revolution= 251rad/sec
To find the acceleration of a point on the edge of the grinding wheel, we can use the formula for centripetal acceleration:
a = (v^2) / r
Where:
a is the centripetal acceleration,
v is the linear velocity of the point on the edge, and
r is the radius of the grinding wheel.
First, let's calculate the linear velocity (v) of a point on the edge of the grinding wheel. The linear velocity can be determined by multiplying the angular velocity (ω) with the radius (r).
The formula for angular velocity is:
ω = (2π * n) / t
Where:
ω is the angular velocity,
π is a mathematical constant (approximately 3.14159),
n is the number of revolutions per minute (rpm), and
t is the number of seconds per minute.
Converting the rpm to radians per second (rad/s), we get:
ω = (2π * 2400) / 60
Calculating this gives us an angular velocity of:
ω = 80π rad/s
Now, let's calculate the radius (r) of the grinding wheel. We are given the diameter (d) of the grinding wheel, which can be converted to radius by dividing it by 2.
Given diameter (d) = 0.28 m,
Radius (r) = d / 2 = 0.28 / 2 = 0.14 m
Finally, we can substitute the values of angular velocity (ω) and radius (r) into the centripetal acceleration formula:
a = (v^2) / r
a = (ω^2 * r) / r
a = ω^2
Substituting ω = 80π rad/s into the formula gives us the final result:
a = (80π rad/s)^2
a ≈ 20,106.19 m/s^2
Therefore, the acceleration of a point on the edge of the grinding wheel is approximately 20,106.19 m/s^2.
To find the acceleration of a point on the edge of the grinding wheel, we can use the formula for centripetal acceleration:
a = r * ω²
where:
a is the acceleration
r is the radius of the grinding wheel
ω is the angular velocity of the grinding wheel
Given that the diameter of the grinding wheel is 0.28 m, we can calculate the radius:
r = d/2 = 0.28/2 = 0.14 m
The angular velocity is given as 2400 rpm. To convert it to radians per second, we need to multiply it by 2π/60, since there are 2π radians in a full revolution and there are 60 seconds in a minute:
ω = (2400 rpm) * (2π rad/1 rev) * (1 rev/60 s) = 80π rad/s
Now we can substitute the values into the formula to find the acceleration:
a = (0.14 m) * (80π rad/s)²
≈ (0.14 m) * (6400π² rad²/s²)
≈ 2816π² m/s²
Therefore, the acceleration of a point on the edge of the grinding wheel is approximately 2816π² m/s².