As noted in the text, the scalar product between two vectors A and B can becalculated using trigonometry as:Ả B = \Ä||B| cos 0and the vector product as:ẢxB = |Ã||B| sin 0 fAIn words, the scalar product between vectors A and B is the product of theirmagnitudes and the cosine of the angle between them. The vector productbetween vectors A and B is the product of their magnitudes and the sine of theangle between them. The direction of the resulting vector is perpendicular tothe plane containing the vectors, with the sign of the vector following the right-hand rule.The right-hand rule is as follows: begin at vector A and curl the fingers of yourright hand toward vector B. The direction of your thumb is the direction of theresulting vector. In the traditional sense, the positive direction is above ahorizontal plane and the negative direction is below it.These quantities can also be calculated if the vectors are expressed in (i,j,k)notation. IfA = Ayî + Ayƒ + A,kandB = B,1 + B,} + B,kthenÃ·B = A,B, + A,By + A,B,andAxB = \A Ay Az =(A,B, - A,B2)î + (A,B, – A,Bx)} + (A, B,- Ay Bx)k|Bx By Bz53% Both the scalar and vector product have applications in mechanics. Thedisplacement vector and force vector can be multiplied by both methods,resulting in different physical quantities.If multiplied as a scalar product, the resulting quantity is the work done whilepushing or pulling an object from one location to another:W = d = Fd cos 0If multiplied as a scalar product, the result is the torque acting on the object,causing it to begin spinning about an axis, or change its rate of spin:T= ixF = Fr sin 0 înWhen considering work done, the displacement is usually symbolized with a d'(as in scalar distance) whereas with torque, the displacement is usuallysymbolized with an 'r' (as in the radius of a spinning circle).In this challenge exercise, we are asking you to confirm that both methods ofcalculating the scalar and vector products.Let a displacement vector in the (X,Y) plane be 5.0 m at a 30 degree angle withrespect to the x-axis and the force vector 6.0 N at a -45 degree angle withrespect to the x-axis.Calculate the work done by the force in moving an object that distance usingboth methods (trigonometry and vector components).Then calculate the torque acting at that displacement by that force about theorigin of the axis system, including the direction of the torque. (Hint: it will beeither the +Z or -Z direction.)Your solutions should show all the steps necessary to determine the finalanswers.53%

Given , displacement vector is 5m at a 30 degree with respect to x axis. force vector is…

Let a displacement vector in the (X,Y) plane be 5.0 m at a 30 degree angle with respect to the x-axis and the force vector 6.0 N at a -45 degree angle with respect to the x-axis. Calculate the work done by the force in moving an object that distance using both methods (trigonometry and vector components). Then calculate the torque acting at that displacement by that force about the origin of the axis system, including the direction of the torque. (Hint: it will be

Let a displacement vector in the (X,Y) plane be 5.0 m at a 30 degree angle with respect to the x-axis and the force vector 6.0 N at a -45 degree angle with respect to the x-axis. Calculate the work done by the force in moving an object that distance using both methods (trigonometry and vector components). Then calculate the torque acting at that displacement by that force about the origin of the axis system, including the direction of the torque. (Hint: it will be

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

See similar textbooks

Similar questions

A horizontal 810-N merry-go-round of radius 1.70 m is started from rest by a constant horizontal force of 45 N applied tangentially to the merry-go-round. Find the kinetic energy of the merry-go-round after 4.0 s. (Assume it is a solid cylinder. Also assume the force is applied at the outside edge.) Need Help? Read It
acts on a pebble with position vector 7 = (1.92 m ) – (5.98 m )ê, re ), relative to the origin. What is the resulting torque acting on the pebble about (a) the origin and (b) a point with coordinates (8.32 m, 0, -7.80 m)? Force F = (4.70 N ) - (5.35 Nac (a) Number (b) Number i + i + k Units k Units
(a) When opening a door, you push on it perpendicularly with a force of 38.0 N at a distance of 0.830 m from the hinges. What torque (in N-m) are you exerting relative to the hinges? (Enter the magnitude.) N•m (b) Does it matter if you push at the same height as the hinges? O Yes O No
Consider the force vector F=<1,1,1/2>. If the magnitude of the torque T=OP*F is equal to the area of the equilateral triangle formed by the origin, P, and (1,-1,1), then determine the acute angle formed by OP and F. Give your answer in degrees.
A bowling ball encounters a 0.760-m vertical rise on the way back to the ball rack, as the drawing illustrates. Ignore frictional losses and assume that the mass of the ball is distributed uniformly. The translational speed of the ball is 6.78 m/s at the bottom of the rise. Find the translational speed at the top.
In the construction of railroads, curvature of the track is measured in the following way. First a 100.0-ft-long chord is measured. Then the curvature is reported as the angle subtended by two radii at the endpoints of the chord. (The angle is measured by determining the angle between two tangents 100.0 ft apart; since each tangent is perpendicular to a radius, the angles are the same.) In modern railroad construction, track curvature is kept below 1.20°. What is the radius of curvature of a “1.20° curve”? answer in ft
The radius of a wheel is 0.890 m. A rope is wound around the outer rim of the wheel. The rope is pulled with a force of magnitude 5.00 N, unwinding the rope and making the wheel spin CCW about its central axis. Ignore the mass of the rope. (a) How much rope unwinds while the wheel makes 1.00 revolution? ___m (b) How much work is done by the rope on the wheel during 1.00 revolution? ___J (c) What is the torque on the wheel about its axis due to the rope? ____N⋅m
A force of magnitude 30 lbs acts on the end of a wrench as shown. The wrench is 8 inches long and makes an angle of 30° from the -x axis in the x-y plane as shown. What is the torque exerted by this force around point P? Give the magnitude and direction as a vector.
Force F = (- 5.5 N )î + (2.9 N )ĵ acts on a particle with position vector 7 magnitude of the torque on the particle about the origin and (b) the angle between the directions of 7 and F ? = (4.3 m )î + (8.3 m )§.What are (a) the
Force F = (8.55 N )î – (9.95 N )ê acts on a pebble with position vector T (6.87 m )ĵ – (2.53 m )k, relative to the origin. What is the resulting torque acting on the pebble about (a) the origin and (b) a point with coordinates (8.02 m, 0, - 6.52 m)?
1 Force F = (7.26 N )i^— (7.05 N )ê acts on a pebble with position vector ☞ = (5.76 m )j^– (3.10 m), relative to the origin. What is the resulting torque acting on the pebble about (a) the origin and (b) a point with coordinates (5.00 m, 0, -5.54 m)? - Your answer is partially correct. (a) Number www (b) Number Hint eTextbook and Media Save for Later -40.608 52 -40.608 80 F3 Q F4 + + JUL 2 i 22.51 9 F5 -18.26 tv MacBook Air C F6 + ← F7 -41.817 -41.82 ST DII F8 k Units N-m Attempts: 2 of 4 used A kUnits F9 N-m Submit Answer A F10 + # F11
Force F = (-7.0 N)î + (5.0 N) ĵ acts on a particle with position vector7 = (4.0 m)î + (5.0 m) ĵ. (a) What is the torque on the particle about the origin, in unit-vector notation? = N: m (b) What is the angle between the directions of r and Torque is the cross product of a position vector (extending from a chosen point, here the origin, to the particle) and a force vector. Did you take the cross product in unit-vector notation? Do you remember how find the angle between two vectors by taking a dot product in both unit-vector notation and also in magnitude-angle notation? (You can similarly use a cross product to do this.) Do you remember how to find the magnitude of a vector from its components?