A rectangular plate has width and breadth of 300 cm and 400...

A rectangular plate has width and breadth of 300 cm and 400 cm, respectively. An applied torque of 100 N.m gives the plate an angular acceleration of 10 rad/s2 about a perpendicular axis through the centre of the plate. 1) Calculate the rotational inertia of the plate about that axis. 2) Determine the mass of the plate. 3) Suggest the radius of a solid sphere that has the same mass and rotational inertia as that of the plate.

Transcript

00:01 We have a square plate here with a side of 0 .18 meters and it's acted upon by three forces as shown in the figure.

00:08 The axis of rotation or the pivot passes through the center of the plate as also shown.

00:16 Okay, there are three requirements in the problem.

00:19 We are to determine the net torque acting on the plate with reference to the center.

00:27 And then the angular acceleration and the final angular velocity after it has rotated for three seconds.

00:36 We will be using the definition of torque here, which is a product of the applied force and the moment arm.

00:44 Moment arm is the perpendicular distance between the pivot and the line of action of an applied force.

00:52 For the sign, if the applied force makes the beam rotate counterclockwise, we'll take it as positive torque and negative if otherwise.

01:03 Okay.

01:04 So for the first part, net torque about the axis here, the individual applied force is given.

01:19 So we just need to determine the moment arm with respect to the pivot.

01:24 So let's have it here.

01:27 Let's start with force one.

01:29 Force one and force two are downward.

01:32 So the line of action is along this side.

01:35 So from the pivot towards the line of action, this horizontal line now is perpendicular to the line of action of f1.

01:44 Therefore, from the center to the edge of the plate, this is now the moment arm of force.

01:50 One.

01:51 Same explanation for force two.

01:54 This is the line of action.

01:56 And this horizontal line makes a 90 degree angle with the line of action, hence this distance is the moment arm of force two.

02:06 Now, since force three makes a 45 degree angle with a horizontal side here of the plate, then from the center, along this diagonal, let me write it properly.

02:19 Oops.

02:20 Should be straight.

02:24 Okay, wait, wait, wait, wait, last time.

02:27 Okay, let's just assume that this is straight.

02:29 So this diagonal line now, if we extrapolate this, this makes a 90 -degree angle with this line of action of force three.

02:38 Hence, this diagonal line is the moment arm of force three.

02:42 So let's list the moment arms now.

02:45 The moment arm of force one is just half of 0 .18, so it's 0 .090 meter, which is also equivalent to the moment arm of force 2.

02:55 And then the moment arm of force 3, we just use pythagorean relation.

02:59 So that for this triangle here, the base and the height are equal.

03:06 So we have 0 .09.

03:09 We square this.