A centrifuge consists of four cylindrical containers, each...

A centrifuge consists of four cylindrical containers, each of mass $m$, at a radial distance $r$ from the rotation axis. Determine the time $t$ required to bring the centrifuge to an angular velocity $\omega$ from rest under a constant torque $M$ applied to the shaft. The diameter of each container is small compared with $r,$ and the mass of the shaft and supporting arms is small compared with $m$.

Key Concepts

Torque

Torque is the rotational equivalent of force. It is a measure of the tendency of a force to rotate an object about an axis or pivot. In rotational dynamics, a constant torque applied to a system results in a constant angular acceleration, analogous to applying a constant force producing linear acceleration in translational motion.

Moment of Inertia

The moment of inertia quantifies how a body's mass is distributed with respect to a chosen axis of rotation, acting as the rotational analogue of mass in linear kinematics. It determines the resistance of an object to changes in its rotational state, with larger moments of inertia indicating that more torque is needed to achieve the same angular acceleration.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time. In rotational dynamics, when a constant torque is applied, the angular acceleration remains constant, and it is directly obtained by dividing the torque by the moment of inertia, as expressed in Newton's second law for rotation.

Rotational Kinematics

Rotational kinematics encompasses the equations that relate angular displacement, angular velocity, angular acceleration, and time, analogous to linear kinematics. Under constant angular acceleration, these equations allow us to determine the time required to reach a certain angular velocity from rest, with the relationship that the final angular velocity equals the product of angular acceleration and time.

Transcript

00:01 So here on this problem we are given this centrifuge that consists of four cylinder as you can see here and the mass of each cylinder is small m and the radial distance from the center of rotation is the center of rotation and this distance is given as r now initially this centrifuge was at rest now we have to calculate the time required to bring the centrifuge to hang on a velocity of omega and by applying a constant torque capital m.

00:31 Now to solve this problem, we know that the net talk or the net moment acting on the body is equals to the rate of change of angular momentum.

00:41 That is we can write that m .0 is equals to dh over d t or we can write that sigma m0 is equals to h0 dot.

00:58 It's the another way to write this form.

01:02 Now we can take d t to the left side so m not d t is equals to d h we can integrate both the side now we have to calculate the time t so initially it was at zero and the angular momentum was zero when the time becomes t the angular becomes the angular becomes h0 so it would be equals to this thing would become m0 times t is equal to h where h0 is the final angular momentum when the angular velocity of the centrifuge fuse becomes omega now from the concept of angular momentum we know that angular momentum h0 is equals to the cross product of the position vector of masses and the linear momentum m i vi and in this case h not would be equals to position vector is r and there are four masses so four m and the velocity can be written as r omega so it would be r omega so h not comes out to be as this is 4 m r square omega so i can plug it here and m not is the moment or external moment acting on the body which is in this case it is m so we can write this to be m times t is equals to h not we have 4 m r square times omega now we have to calculate the expression for time t...

Please give Ace some feedback