A square metal plate 0.180 m on each side is pivoted about...

A square metal plate 0.180 m on each side is pivoted about an axis through point $O$ at its center and perpendicular to the plate (Fig. $\mathbf{E 1 0 . 3}$ ). Calculate the net torque about this axis due to the three forces shown in the figure if the magnitudes of the forces are $F_{1}=18.0 \mathrm{~N}, F_{2}=26.0 \mathrm{~N},$ and $F_{3}=14.0 \mathrm{~N}$. The plate and all forces are in the plane of the page.

Key Concepts

Sign Conventions in Torque

Sign conventions in torque involve assigning positive or negative values to torques based on their direction (e.g., clockwise or counter-clockwise). Consistent use of these conventions is essential for correctly summing torques and analyzing the rotational motion of a system.

Net Torque

Net torque is the sum of all individual torques acting about the same axis, taking into account their rotational directions. It quantifies the overall rotational effect on an object and determines whether the object will experience angular acceleration.

Lever Arm

The lever arm is the perpendicular distance from the axis of rotation to the line along which a force acts. It plays a critical role in calculating torque since a longer lever arm increases the torque produced by a given force.

Torque

Torque is the measure of the rotational influence of a force acting at a distance from a pivot. It is given by the product of the force magnitude and the perpendicular distance (lever arm) from the axis of rotation to the force's line of action, determining the tendency of the force to rotate the object.

Transcript

00:01 This problem is asking us about three forces on a square metal plate that is free to rotate about an axis going through its center.

00:11 F1 is given to us at 18 newtons.

00:14 F2 appears given to us as 26 newtons.

00:18 And f3 down here is given to us as 14 newtons going at a 45 degree angle from the plate side.

00:27 And the length of each side of the plate is 0 .18 meters.

00:31 So using all that in our torque formula, we're supposed to find the total torque on this plate, which the total torque is just going to be equal to each of the individual torques added together.

00:49 So now all we need to do is find each of these torques.

00:53 Torc one, plug in our torque formula, r1, f1, sine, phi, 1.

01:04 And we have f1 already, so all we need to need to.

01:07 To do is find r1 and phi 1 first off r1 we can just use the diagonal through the square that will give us the total distance from the center where it's pivoting so all we need to do is use the pathagorean theorem now that this can be square root of 2 times l over 2 so that'll be 0 .09 times a square root of 2 which will get us a an r value of 0 .127.

01:58 And then all we need now is our phi.

02:06 Now we can't use these right angles, of course, because this phi is not from the side of the square.

02:13 It'll be from the radius to the center.

02:18 So since we know that these were right angles, and since we know, of course, that the diagonal square goes up at 45 degrees, we can just add them together and get this is going to be 135 degrees...

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