The ultracentrifuge is an important tool for separating and analyzing proteins in biological research. Because of the enormous centripetal accelerations that can be achieved, the apparatus must be carefully balanced so that each sample is matched by another on the opposite side of the rotor shaft. Failure to do so is a costly mistake. Any difference in mass of the opposing samples will cause a net force in the horizontal plane on the shaft of the rotor. Suppose that a scientist makes a slight error in sample preparation, and one sample has a mass 11mg greater than the opposing sample.
If the samples are 11cm from the axis if the rotor and the ultracentrifuge spins at 7.1×10^4 rpm, what is the magnitude of the net force on the rotor due to the unbalanced samples?
First, we need to convert the rotation speed to rad/s. Given that 1 rotation is equal to 2π radians, and 1 minute has 60 seconds, we have:
(7.1 × 10^4 rotations/min) × (2π radians/rotation) × (1 min/60 s) = 7.47 × 10^3 radians/s
Next, we can compute the centripetal acceleration for the unbalanced mass:
a = ω^2 × r
a = (7.47 × 10^3)^2 × 0.11
a ≈ 3.71 × 10^7 m/s^2
Now, we can calculate the net force acting on the unbalanced mass:
F = m × a
F = (11 × 10^-6 kg) × (3.71 × 10^7 m/s^2)
F ≈ 0.408 N
The magnitude of the net force on the rotor due to the unbalanced samples is approximately 0.408 N.
To find the magnitude of the net force on the rotor due to the unbalanced samples, we can use the concept of centripetal force.
1. Convert the given spinning speed from rpm to radians per second:
- 7.1×10^4 rpm = (7.1×10^4 rpm) * (2π rad/1 min) * (1 min/60 s)
- This gives the spinning speed as approximately 7433.87 radians per second.
2. Convert the given distance from the axis of the rotor from centimeters to meters:
- 11 cm = 0.11 meters
3. Calculate the net force using the equation for centripetal force:
- Centripetal Force (F) = mass (m) * angular velocity squared (ω^2) / radius (r)
- In this case, the mass difference between the samples is 11 mg, which can be converted to kilograms by dividing by 1000:
- Mass difference (m) = 11 mg = 11 * 10^(-6) kg
- The angular velocity squared (ω^2) can be calculated by multiplying the spinning speed by itself:
- Angular velocity squared (ω^2) = (7.1×10^4 rad/s)^2
- The radius (r) is the distance from the axis, which is 0.11 meters.
- Plugging in the values:
- F = (11 * 10^(-6) kg) * ((7.1×10^4 rad/s)^2) / 0.11 meters
Calculating this equation will give you the magnitude of the net force on the rotor due to the unbalanced samples.
To find the magnitude of the net force on the rotor due to the unbalanced samples, we need to consider the centripetal force acting on each sample.
The centripetal force is given by the equation:
F = (m * v^2) / r
where F is the force, m is the mass of the sample, v is the linear velocity, and r is the radius.
First, we need to find the linear velocity of the samples. Since the centrifuge spins at 7.1×10^4 rpm, we need to convert this value to rad/s:
Linear velocity (v) = (Angular velocity (ω) * radius (r))
The angular velocity can be determined by converting the rpm (revolutions per minute) to rad/s:
Angular velocity (ω) = (7.1×10^4 rpm) * (2π rad / 60 s)
Next, we can determine the linear velocity using the given radius (r = 11 cm = 0.11 m):
Linear velocity (v) = (7.1×10^4 rpm) * (2π rad / 60 s) * 0.11 m
Now that we have the linear velocity, we can calculate the centripetal force exerted on each sample:
F = (m * v^2) / r
For the unbalanced sample, its mass is 11 mg greater than the opposing sample. Let's denote the mass of the opposing sample as m_opposing, and the mass of the unbalanced sample as m_unbalanced. Therefore:
m_unbalanced = m_opposing + 11 mg
Since mass is given in milligrams, we need to convert it to kilograms:
m_unbalanced = (m_opposing + 11 × 10^-6 kg)
Now, we can calculate the centripetal force on each sample using the linear velocity and radius:
F_opposing = (m_opposing * v^2) / r
F_unbalanced = (m_unbalanced * v^2) / r
To find the net force caused by the unbalanced samples, we need to subtract the force exerted by the opposing sample from the force exerted by the unbalanced sample:
Net force = |F_unbalanced - F_opposing|
By substituting the values into the equations and performing the calculations, we can find the magnitude of the net force caused by the unbalanced samples.