Torque is a very important quantity in physics, specially while dealing with rotational motion. It is...

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Torque is a very important quantity in physics, specially while dealing with rotational motion. It is denoted by 7 and is defined as the cross product of F and F (in the order), where is the position vector of the point where the force F is being applied, relative to some point about which torque is being measured. Consider, F = 31 - 43+k F = mi + 12j - 3k where i, j,k are unit vector along x, y and z axes respectively. (a) Find the value of m for which 7 = 0. (b) If m = 3, then find the torque 7 and calculate the angles which the torque 7 makes with r, y and z axis. (c) Now, consider the projection of and F on the zy plane such that, F = i-2j F = 3i+6j If and are the unit vector along z' and y' in a rotated coordinate system as shown in the figure, +y (i) Show that, L +X rotation by angle i= cos 0i+ sin 0j 3 = -sin 0i + cos 0} (ii) If 0 = 30, then show that 7. F is conserved in both coordinate system that is - 7. F=F.F 2 3 2 5 Hints: (i) The vectors in both coordinate systems will have the same magnitude. (ii) You might use the results of question (i) to find the relation between the components of vectors in primed and unprimed coordinates. Torque is a very important quantity in physics, specially while dealing with rotational motion. It is denoted by 7 and is defined as the cross product of F and F (in the order), where is the position vector of the point where the force F is being applied, relative to some point about which torque is being measured. Consider, F = 31 - 43+k F = mi + 12j - 3k where i, j,k are unit vector along x, y and z axes respectively. (a) Find the value of m for which 7 = 0. (b) If m = 3, then find the torque 7 and calculate the angles which the torque 7 makes with r, y and z axis. (c) Now, consider the projection of and F on the zy plane such that, F = i-2j F = 3i+6j If and are the unit vector along z' and y' in a rotated coordinate system as shown in the figure, +y (i) Show that, L +X rotation by angle i= cos 0i+ sin 0j 3 = -sin 0i + cos 0} (ii) If 0 = 30, then show that 7. F is conserved in both coordinate system that is - 7. F=F.F 2 3 2 5 Hints: (i) The vectors in both coordinate systems will have the same magnitude. (ii) You might use the results of question (i) to find the relation between the components of vectors in primed and unprimed coordinates.