If the aluminum cube of Prob. 18.21 were replaced by a cube...

Key Concepts

Moment of Inertia

The moment of inertia is a measure of how a body's mass is distributed relative to the axis around which it rotates. It determines how resistant the body is to changes in its rotational motion. A key factor in calculating rotational dynamics, the moment of inertia dictates the relationship between applied torque and the resulting angular acceleration.

Torque

Torque is the rotational equivalent of force and represents the ability of a force to cause an object to rotate about an axis. It is calculated by multiplying the force by the lever arm (the distance perpendicular to the direction of force from the pivot). In rotational dynamics, torque is the primary quantity that causes angular acceleration.

Angular Acceleration

Angular acceleration is the rate at which an object’s angular velocity changes with time. It arises when a net torque is applied to a body and is inversely proportional to the moment of inertia. This relationship is central when determining how quickly an object can begin to rotate from rest under an applied force.

Rotational Kinematics

Rotational kinematics deals with the motion of rotating bodies using analogs of the equations for linear motion. It includes concepts such as angular displacement, angular velocity, and angular acceleration, and is used to determine quantities like the time period required for a full rotation under constant acceleration.

Rigid Body Dynamics

Rigid body dynamics is the study of the motion of bodies in which the distance between any two points remains constant, meaning the body does not deform under the applied forces. This field combines concepts such as torque, moment of inertia, and angular acceleration to analyze and predict the rotational and translational behavior of objects when forces are applied.

Transcript

00:01 We have this cube here which is pushed with the force at the point c, and we define the time that it takes for this cube to complete one full revolution, knowing that it can rotate about this diagonal from point a to point b.

00:18 And we know that the mass, so the mass, mass m prime, this is the mass of one side of this cube, which we can get by taking the total mass and dividing it by six.

00:33 And the mass of one side of the cube is given as 8 kilograms.

00:43 So 8 kilograms.

00:49 And when we're solving this problem, we're going to assume that the x, y, and z axis are all perpendicular to the faces of this cube.

01:01 So we're going to start by finding what the moment of inertia is for a side that's perpendicular to the x axis.

01:11 So we'll call that i of x1.

01:13 So i of x1, and we can write that as 1 over 6 times m prime times a square.

01:29 And next we'll write the moment of inertia for a side that's perpendicular to the y or z axis with respect to the x axis.

01:42 So we'll call that i of x2, and that's going to be equal to 1 over 12 plus 1 over 4 times.

01:54 M prime times a squared and we can simplify this as 1 over 3 times m prime times a squared so now to get the moment of inertia about the x -axis we can take these two terms here and take the summation of them so to write that i of x we're going to write that as two times i of x one since there are two sides that are perpendicular to the x -axis, and then plus four times i of x2, since there's four sides that are perpendicular to either the y or z -axis.

02:50 So now we can just plug in those values.

02:56 And we're going to write that as 1 over 6 times m over 6 times a squared, plus 4 times 1 over 3 times m over 6 times a squared.

03:15 And we can simplify this to 5 over 18 times m times a squared...

Please give Ace some feedback