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Numerade Educator

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Problem 41 Hard Difficulty

A wrench 30 cm long lies along the positive y-axis and grips a bolt at the origin. A force is applied in the direction $ \langle 0, 3, -4 \rangle $ at the end of the wrench. Find the magnitude of the force needed to supply $ 100 N \cdot m $ of torque to the bolt.

Answer

$|\mathbf{F}|=\frac{1250}{3} \approx 417$

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Video Transcript

Hello. I have a question. It is given that Parents 30 Centim long. So uh length is 30 centimetre lies along the positive Y axis. And groups boy at the origin force is applied in the direction zero comma three for minus four at the end of the range. So we have to find the marriage of the force needed to supply and then you to meter. So dark is being given as handed. Newton meters and rent will be the radius definitely it will be the radius of the rotation therapies and images. That is 0.30 m. And force the applied in the direction zero comma three comma. Mine is full. So it is just like this is why here is the board and wrenches lying like this. This is our and the forces applied in the direction zero comma three comma minus four. Remember this is only the direction not the force. So let's suppose in any direction this force is being applied. Okay. Our accent why both are positive solid is supposed forces being applied in this direction. Okay. We know that dark equal to ask victor's cross affected. Okay. Darkies are victor across affected and if we just take the magnitude it becomes equal to talk equal tours that affected as magnitude of ours are let us say into earth and scientists. So basically we need to find the value tita for that we'll be using the fact that the direction of this are victor is direction of this. Our victories zero comma one comma zero as it is lying along the y exist. Okay. And little camera And not 10 comma are because it is a lentil, Alexis are 0.3 m. Why is 0.3 m and zero. And we know that the direction of force is being given as zero comma £3 minus four zero comma three comma minus four. So if we make it dot product that it got ripped off victor expected it becomes equal to magnitude of our victors, magnitude of a vector cost. Uh Cause teacher Okay so this is we are doing for the only for the direction, we are doing only for the direction these things after it's not actual force. What direction is it will be corporate of these two which will become zero plus 3 to 0.3. Place zero This magnitude will be zero, press zero point 09 into nine plus 16. And car seat to Okay this is three and 2 0.3 Equal to 0.3. Mhm Little pointy Into this will be 25. So five cost data. These two will get cancelled out. So course it is three x 5 and of course it a three x 5 according to Pythagorean identity. Scientists will be one nine x 25. Before by five. Now let us plug in over here in the formula formula for talk. So talk will be asked f scientist to which is followed by a file. So f will be equal to five and 2 Talk by foreign to us. So five into targets 100 m m by four into, uh, is 30 cm, so 0.30. This will be 25, so 125 by 0. 1 Thing, you know, 25 416 points 67 million. So they should be required force. Thank you.