(b)A centrifuge has a maximum rotation rate of 10,000 rpm and can be stopped in 4 seconds. Assume the deceleration is uniform. The centrifuge radius is 8 cm.
What is the distance that a point on the rim travels during deceleration?
To find the distance that a point on the rim of the centrifuge travels during deceleration, we need to use the formula for linear distance traveled during uniform acceleration:
S = (v^2 - u^2) / (2a)
Where:
S is the distance traveled
v is the final velocity
u is the initial velocity
a is the acceleration
In this case, the initial velocity is the maximum rotation rate of the centrifuge, which is given in revolutions per minute (rpm). We need to convert this to radians per second (rad/s).
1 revolution = 2π radians
1 minute = 60 seconds
So, the initial velocity (u) is:
u = 10,000 rpm * 2π rad/rev * (1 min / 60 s) = 1047.2 rad/s
The final velocity (v) is 0, as the centrifuge comes to a stop.
The acceleration (a) can be calculated using the formula:
a = (v - u) / t
Where:
v is the final velocity (0 m/s)
u is the initial velocity (1047.2 rad/s)
t is the time taken to stop (4 s)
a = (0 - 1047.2 rad/s) / 4 s = -261.8 rad/s^2 (negative because it's decelerating)
Now we can substitute these values into the formula for distance:
S = (v^2 - u^2) / (2a) = (0 - (1047.2 rad/s)^2) / (2 * -261.8 rad/s^2)
Simplifying further,
S = (-1095302.24 rad^2/s^2) / (-523.6 rad/s^2)
S = 2092.31 rad
Finally, we need to convert the distance from radians to centimeters. Since the radius of the centrifuge is 8 cm,
Distance in cm = 2092.31 rad * 8 cm/rad = 16,738.5 cm
Therefore, a point on the rim of the centrifuge travels approximately 16,738.5 cm during deceleration.
To calculate the distance that a point on the rim of the centrifuge travels during deceleration, we can use the formula:
Distance = (initial velocity * time) + (0.5 * acceleration * time^2)
First, let's convert the maximum rotation rate of 10,000 revolutions per minute (rpm) to radians per second (rad/s). Since 1 revolution is equal to 2π radians, we can calculate the angular velocity as follows:
Angular velocity = (10,000 rpm) * (2π rad/1 min) * (1 min/60 s) = (10,000 * 2π) / 60 rad/s
Now, let's calculate the initial velocity. Since the initial velocity is the same as the maximum angular velocity, we have:
Initial velocity = (10,000 * 2π) / 60 rad/s = (10,000 * π) / 30 rad/s
Next, let's calculate the deceleration. Deceleration is defined as the negative acceleration. Since the deceleration is uniform, it is equal to the acceleration during deceleration:
Deceleration = - (initial velocity / time) = - ((10,000 * π) / 30) / 4 rad/s^2
Now, substituting these values into the distance formula, we get:
Distance = ((10,000 * π) / 30) * 4 + 0.5 * (- ((10,000 * π) / 30) / 4) * 4^2
Simplifying this equation will give us the distance that a point on the rim travels during deceleration.