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Tightening a bolt Suppose you apply a force of $20 \mathrm{N}$ to a 0.25 -meter-long wrench attached to a bolt in a direction perpendicular to the bolt. Determine the magnitude of the torque when the force is applied at an angle of $45^{\circ}$ to the wrench.

The magnitude of the torque is $\frac{5}{\sqrt{2}}N\cdot m$.

Calculus 3

Chapter 13

Vectors and the Geometry of Space

Section 4

Cross Products

Vectors

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Hello, everybody. In this video, we'll be showing you how to solve Exercise 49 chapter 13 Section four of calculus, early insurance and dental. Now, in this problem, they tell us that we have a bolt here that goes into the page and perpendicular to it with the page, we have a wrench that is 0.25 meters long and at the end of this ridge of force of 20 Newtons is applied at an angle of 45 degrees. And with this they want us to determine the magnitude of the tour here at the bolt produced by this force on this red. Not to do this, we're going to use the euro 13.3, which states the magnitude of the cross product of two vectors, such as or cross F, which equals the torch, is equal to the magnitude of, or friends and magnitude of f just the product of the magnitudes of both factors times the sign of their angle of separation now, or is a vector that extends from the pivot point to the place where the force is applied as drawn. And so we know that, or has a magnitude of 0.25 meters just the length of the wrench. And we know that F has a magnitude of 20 Newtons and then we also know that as or is along the wrench, that f is applied at a 45 degree angle to our so Therefore, we also know data equals 45 degrees. And with that we can directly calculate this magnitude we have that the magnitude of this torque is equal 0.25 meters times 20 Newton times the sign of 45 degrees. This is equal 0.25 times 20 which is just five times one over route to Newton meters. So we see. But the magnitude of this torque is equal to five, divided by the square root of two Newton meters. And this is our answer. The problem. So that's how you solve exercise 49

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