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297Chap3

# 297Chap3 - External and internal forces External forces The...

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External and internal forces External forces The action of other bodies on the rigid body under con- sideration. Internal forces The forces which hold together the particles forming the rigid body.

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CE 297 2 Principle of transmissibility A force may be considered to act at any point along its line of action so far as its effect on a rigid body is concerned. A force on a rigid body may be treated as a sliding vector.
CE 297 3 Vector (cross) product of two vectors Result is a vector V = P × Q The line of action of V is perpendicu- lar to the plane of P and Q The magnitude of V is given by V = PQ sin θ 0 < θ < 180° The direction of V is given by the right-hand rule.

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CE 297 4 Vector (cross) product of two vectors (cont.) Properties of the vector product Not commutative P × Q = - ( Q × P ) Distributive P × ( Q 1 + Q 2 ) = P × Q 1 + P × Q 2 Not associative ( P × Q ) × S P × ( Q × S ) Magnitude of V equals the area of the parallelogram defined by P and Q .
CE 297 5 Vector (cross) product of two vectors (cont.) Unit vectors i × i = 0 j × i = - k k × i = j i × j = k j × j = 0 k × j = - i i × k = - j j × k = i k × k = 0

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CE 297 6 Vector (cross) product of two vectors (cont.) In terms of rectangular components, V = P × Q = ( P x i + P y j + P z k ) × ( Q x i + Q y j + Q z k ) = P x Q x ( i × i ) + P x Q y ( i × j ) + P x Q z ( i × k ) P y Q x ( j × i ) + P y Q y ( j × j ) + P y Q z ( j × k ) P z Q x ( k × i ) + P z Q y ( k × j ) + P z Q z ( k × k ) = ( P y Q z - P z Q y ) i + ( P z Q x - P x Q z ) j + ( P x Q y - P y Q x ) k
CE 297 7 Vector (cross) product of two vectors (cont.) V = P × Q = ( P y Q z - P z Q y ) i + ( P z Q x - P x Q z ) j + ( P x Q y - P y Q x ) k V x = P y Q z - P z Q y V y = P z Q x - P x Q z V z = P x Q y - P y Q x V = i P x Q x j P y Q y k P z Q z Add and subtract the diagonal products of the matrix + i P x Q x + j P y Q y + - k P z Q z - i P x Q x - j P y Q y

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CE 297 8 Moment of a force about a point Moment of F about point O M O = r × F The line of action of M O is per- pendicular to the plane of r and F The magnitude of M O is M 0 = rF sin θ = F ( r sin θ ) = Fd The direction of M O is given by the right-hand rule. M O measures the tendency of F to rotate the rigid body about an axis directed along M O
CE 297 9 Moment of a force about a point (cont.) For two dimensional problems, The line of action of M O is perpendic- ular to the two-dimensional plane The magnitude of M 0 can be repre- sented by a scalar M O = ± Fd The direction is given by a positive (CCW) or negative (CW) sign.

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