Kenneth Kuttler

298 CHAPTER 14. VECTOR PRODUCTS

the position vectors nor the force vectors in the same plane; what then? To keep track ofthis sort of thing, define for each R and F, the torque vector

τ ≡R×F .

This is also called the moment of the force, F . That way, if there are several forces actingat several points the total torque can be obtained by simply adding up the torques associatedwith the different forces and positions.

Example 14.5.2 Suppose R1 = 2i− j+3k,R2 = i+2 j− 6k meters and at the pointsdetermined by these vectors there are forces, F 1 = i−j+2k and F 2 = i−5j+k Newtonsrespectively. Find the total torque about the origin produced by these forces acting at thegiven points.

It is necessary to take R1 ×F 1 +R2 ×F 2. Thus the total torque equals∣∣∣∣∣∣i j k2 −1 31 −1 2

∣∣∣∣∣∣+∣∣∣∣∣∣i j k1 2 −61 −5 1

∣∣∣∣∣∣=−27i−8j−8k Newton meters

Example 14.5.3 Find if possible a single force vector F which if applied at the pointi+j+k will produce the same torque as the above two forces acting at the given points.

This is fairly routine. The problem is to find F = F1i+F2j+F3k which produces theabove torque vector. Therefore,∣∣∣∣∣∣

i j k1 1 1F1 F2 F3

∣∣∣∣∣∣=−27i−8j−8k

which reduces to (F3 −F2) i+ (F1 −F3) j+ (F2 −F1) k= −27i− 8j− 8k. This require-ment amounts to solving the system of three equations in three unknowns, F1,F2, and F3,

F3 −F2 =−27, F1 −F3 =−8, F2 −F1 =−8

However, there is no solution to these three equations. (Why?) Therefore no single forceacting at the point i+j+k will produce the given torque.

14.5.2 Center of MassThe mass of an object is a measure of how much stuff there is in the object. An object hasmass equal to one kilogram, a unit of mass in the metric system, if it would exactly balancea known one kilogram object when placed on a balance. The known object is one kilogramby definition. The mass of an object does not depend on where the balance is used. Itwould be one kilogram on the moon as well as on the earth. The weight of an object issomething else. It is the force exerted on the object by gravity and has magnitude gmwhere g is a constant called the acceleration of gravity. Thus the weight of a one kilogramobject would be different on the moon which has much less gravity, smaller g, than on theearth. An important idea is that of the center of mass. This is the point at which an objectwill balance no matter how it is turned.

298 CHAPTER 14. VECTOR PRODUCTSthe position vectors nor the force vectors in the same plane; what then? To keep track ofthis sort of thing, define for each R and F, the torque vectorT=RxF.This is also called the moment of the force, F’. That way, if there are several forces actingat several points the total torque can be obtained by simply adding up the torques associatedwith the different forces and positions.Example 14.5.2 Suppose R, = 21 -—j+3k,R, =1+2 3-— 6k meters and at the pointsdetermined by these vectors there are forces, F, =t—3+2k and Fy =1—5j +k Newtonsrespectively. Find the total torque about the origin produced by these forces acting at thegiven points.It is necessary to take R, x F, + Ry x F2. Thus the total torque equalsat Jj &k a gj ik2 -1 3 ;+)/ 1 2 £—-6 | =—271—87 —8k Newton meters1 -1l 2 1 -5 1Example 14.5.3 Find if possible a single force vector F' which if applied at the pointt+ 9+k will produce the same torque as the above two forces acting at the given points.This is fairly routine. The problem is to find F = Fjt+ Fog + 3k which produces theabove torque vector. Therefore,ij k1 1 1 |=-27i-8j—8kKR FPwhich reduces to (F3 — Fy) i+ (Fi —F3) 9+ (Fo —F\) k = —27% — 87 — 8k. This require-ment amounts to solving the system of three equations in three unknowns, F), Fo, and F3,Fy Fy = 27, F| -Fy =—-8, h-F, =-8However, there is no solution to these three equations. (Why?) Therefore no single forceacting at the point 2+ 7 + k will produce the given torque.14.5.2 Center of MassThe mass of an object is a measure of how much stuff there is in the object. An object hasmass equal to one kilogram, a unit of mass in the metric system, if it would exactly balancea known one kilogram object when placed on a balance. The known object is one kilogramby definition. The mass of an object does not depend on where the balance is used. Itwould be one kilogram on the moon as well as on the earth. The weight of an object issomething else. It is the force exerted on the object by gravity and has magnitude gmwhere g is a constant called the acceleration of gravity. Thus the weight of a one kilogramobject would be different on the moon which has much less gravity, smaller g, than on theearth. An important idea is that of the center of mass. This is the point at which an objectwill balance no matter how it is turned.