Find the required angular speed, ω, of an ultracentrifuge for the radial acceleration of a point 2.00cm from the axis to equal 6.00×105 g (where g is the acceleration due to gravity).
Well, if you think about it, the question is asking how fast the point needs to spin in order to feel an acceleration 6.00×10^5 times as strong as gravity.
I hope you're ready for some high-speed spinning, because we're about to enter turbo mode!
Now, we know that the radial acceleration, ar, is given by the formula ar = ω²r, where ω is the angular speed and r is the distance from the axis.
In this case, we're given ar = 6.00×10^5g and r = 2.00 cm (or 0.02 m, if you prefer). We need to find ω.
Let's plug in the numbers and solve for ω:
6.00×10^5g = ω²(0.02 m)
Now, since g is the acceleration due to gravity, we can substitute g with 9.8 m/s²:
6.00×10^5(9.8 m/s²) = ω²(0.02 m)
Simplifying further, we get:
5.88×10⁶ = (ω²)(0.02)
To isolate ω², divide both sides by 0.02:
ω² = 5.88×10⁶ / 0.02
ω² = 2.94×10⁸
Finally, take the square root of both sides to find ω:
ω ≈ √(2.94×10⁸)
ω ≈ 17141.4 rad/s
So, there you have it! The required angular speed, ω, for the ultracentrifuge is approximately 17141.4 rad/s. Time to buckle up and prepare for some mind-blowing spinning action!
To find the required angular speed, ω, of the ultracentrifuge, we can use the formula for radial acceleration:
ar = ω^2 * r
where ar is the radial acceleration, ω is the angular speed, and r is the distance from the axis.
We are given the following values:
ar = 6.00×10^5 g
r = 2.00 cm
First, let's convert the distance from centimeters to meters:
r = 2.00 cm = 0.02 m
Next, let's convert the acceleration due to gravity from g to m/s^2:
g = 9.81 m/s^2
Substituting these values into the radial acceleration formula, we get:
6.00×10^5 g = ω^2 * 0.02
Solving for ω, we can rearrange the equation as:
ω^2 = (6.00×10^5 g) / 0.02
ω^2 = 3.00×10^7 g
Taking the square root of both sides, we get:
ω = sqrt(3.00×10^7 g)
Now, substituting the value of g, we have:
ω = sqrt(3.00×10^7 * 9.81)
ω = sqrt(2.943×10^8)
ω ≈ 17137.5 rad/s
Therefore, the required angular speed of the ultracentrifuge is approximately 17137.5 radians per second.
To find the required angular speed, ω, of the ultracentrifuge, we can use the formula for radial acceleration:
a = ω²r
Where:
a is the radial acceleration,
ω is the angular speed,
and r is the distance from the axis.
Here, we are given:
r = 2.00 cm = 0.02 m (after converting cm to m),
a = 6.00 × 10^5 g.
First, let's convert the acceleration from g (acceleration due to gravity) to m/s². The acceleration due to gravity is approximately 9.8 m/s².
a = 6.00 × 10^5 g × 9.8 m/s²/g (canceling out the units of g)
Now, we can solve for ω by rearranging the equation:
ω² = a / r
Substituting the given values:
ω² = (6.00 × 10^5 g × 9.8 m/s²) / 0.02 m
ω² = 2.94 × 10^9 rad/s²
Taking the square root of both sides:
ω ≈ √(2.94 × 10^9 rad/s²)
ω ≈ 54,203 rad/s
Therefore, the required angular speed, ω, of the ultracentrifuge is approximately 54,203 rad/s.
r w^2 = Ac = 6*10^5*9.81
w^2 = 58.9 * 10^5 /.02 = 29.4 *10^7
= 2.94 * 10^8
w = 1.72 * 10^4
= 17,200 radians/sec