A centrifuge is spinning at 3600 rev/min, and when turned off, it rotates 50 times before coming to a stop.
A) What is the initial angular velocity?
B) What is the angle (in radians) that the centrifuge passes through before stopping?
C) What is the angular acceleration?
D) How long does it take for the centrifuge to come to rest?
Show some work. Thanks!
A = 376,991
B=time
1421.22
To find the answers to these questions, we will use the equations of rotational motion.
A) The initial angular velocity (ω_0) can be found using the equation:
ω = ω_0 + αt
We are given the final angular velocity (ω = 0) and the time it takes to stop (t = 50 revolutions). We know that 1 revolution is equal to 2π radians. Therefore, the total angle (θ) covered before stopping is 50 * 2π radians.
Since ω = 0, we can rearrange the equation to solve for ω_0:
0 = ω_0 + αt
Next, we substitute the known values:
0 = ω_0 + α * (50 * 2π radians)
Simplifying, we have:
0 = ω_0 + 100πα radians
Since the centrifuge comes to a stop, it means that the final angular velocity is zero (ω = 0). Substituting ω = 0 into the equation, we can solve for ω_0:
ω_0 = -100πα radians
B) The angle (θ) that the centrifuge passes through before stopping is equal to the area under the velocity-time graph. Since the angular velocity is changing linearly, the graph is a triangle with base (t) and height (ω_0).
The area of a triangle is given by:
Area = 1/2 * base * height
Substituting the values, we have:
θ = 1/2 * t * ω_0
Substituting the known values, we get:
θ = 1/2 * (50 * 2π radians) * (-100πα radians)
Simplifying, we have:
θ = -5000π^2α radians
C) The angular acceleration (α) can be found using the formula:
α = (ω - ω_0) / t
Given that the final angular velocity is zero and the time it takes to stop is 50 revolutions, we can calculate α:
α = (0 - ω_0) / (50 * 2π radians)
Substituting the known values, we get:
α = (-100πα radians) / (50 * 2π radians)
Simplifying, we have:
α = -1/100 radians/second^2
D) To find the time it takes for the centrifuge to come to rest, we can rearrange the equation:
ω = ω_0 + αt
Since the final angular velocity (ω) is zero, we have:
0 = ω_0 + αt
Substituting the known values, we get:
0 = -100πα radians + (-1/100 radians/second^2) * t
Simplifying, we have:
100πα radians = (1/100 radians/second^2) * t
Dividing both sides by (1/100 radians/second^2), we find:
100πα * (100/1) = t
Simplifying, we get:
t = 10000πα seconds
A. ISNT its intial angular velocity given? change it to radians/sec
B. what is 50*2PI
C wf^2=wi^2+ 2*alpha*displacement
solve for alpha.