Answer the following questions about torque. Let 𝐫=O P=10 𝐢 . Which is greater (in magnitude): t

Answer the following questions about torque. Let 𝐫=O P=10 𝐢...

Key Concepts

Torque

Torque is a measure of the tendency of a force to produce rotational motion about an axis or pivot point. It is defined as the cross product of the position vector (from the axis to the point of force application) and the force vector, capturing both the magnitude of the force and its lever arm, which is the perpendicular distance from the line of action of the force to the axis.

Cross Product

The cross product is a vector multiplication operation that results in a vector perpendicular to the plane containing the two original vectors. In the context of torque, the cross product of the position vector and the force vector produces a torque vector whose magnitude is given by the product of the magnitudes of the two vectors multiplied by the sine of the angle between them, highlighting the importance of both the force’s magnitude and its direction relative to the position vector.

Vector Magnitude

The magnitude of a vector represents its length or size. When calculating torque, the magnitude of the resulting vector provides the measure of the rotational effect produced by the force. This magnitude depends on both the distance from the pivot point (the length of the position vector) and the component of the force perpendicular to the position vector.

Transcript

00:01 Hello everybody.

00:03 In this video, i'll be showing you how to solve exercise 47 in chapter 13, section 4 of calculus early transcendentals.

00:11 Now in this problem, we have a rigid body, as shown here on the right, that extends from the point to o to p, where this extension is represented by the vector r equal to 10 times i, which means that this object extends 10 units in the x direction.

00:28 And at the point p, we have two possible forces to consider.

00:32 This one here in blue, f equal to 5i minus 5k, as well as the one in red here, f equal to 4i minus 3j.

00:41 And around this pivot point o here, they want us to consider which of these forces is going to induce a greater magnitude of torque around this point o.

00:57 Now, to answer this question, let's first recall theorem 13 .3 in the textbook.

01:03 This theorem says that the magnitude of, well, yeah, here it'll be the magnitude of torque, which is equal to the magnitude of the cross product of r and f, r cross -f here, is equal to the product of each vector's magnitude times the sign of their angle of separation.

01:31 And so for each of these forces, what we want to do is compute this quantity, magnitude of r times magnitude of f times a sign of the angle that separates r in each respective f and to start let's first note that the magnitude of r for both cases is going to be the same ten as r only has one component in the x direction which is ten so let's start with the first force five i minus five k now the magnitude of f is going to be equal to the square root of five squared plus negative 5 squared, and that's just equal to the square root of 2 times 5 squared, which is equal to 5 root 2.

02:28 And now, if we look at the geometry of the situation, we can see that on the x and z axis, this vector is going to be five units in the x direction and five units in the negative z direction, which means that it will be 45 degrees away from the x -axis, which is where the position vector rests.

02:58 And therefore, for this problem, we may compute that the magnitude of the torque is equal to 10, the magnitude of the position vector, times 5 root 2, times the sign of 45 degrees.

03:17 And this is just equal to 50, 10 times 5, times root 2, times 1 over 5.

03:25 Root 2, these cancel, and therefore the magnitude of this torque from this force is 50.

03:35 So now let's consider the second force, f equals 4i minus 3j...

Please give Ace some feedback