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University Physics Volume 1

Challenge Problems

University Physics Volume 1Challenge Problems

Challenge Problems

88.

Vector B→B→ is 5.0 cm long and vector A→A→ is 4.0 cm long. Find the angle between these two vectors when |A→+B→|=3.0cm|A→+B→|=3.0cm.

89.

What is the component of the force vector G→=(3.0i^+4.0j^+10.0k^)NG→=(3.0i^+4.0j^+10.0k^)N along the force vector H→=(1.0i^+4.0j^)NH→=(1.0i^+4.0j^)N?

90.

The following figure shows a triangle formed by the three vectors A→A→, B→B→, and C→C→. If vector C→′C→′ is drawn between the midpoints of vectors A→A→ and B→B→, show that C→′=C→/2C→′=C→/2.

91.

Distances between points in a plane do not change when a coordinate system is rotated. In other words, the magnitude of a vector is invariant under rotations of the coordinate system. Suppose a coordinate system S is rotated about its origin by angle φφ to become a new coordinate system S′S′, as shown in the following figure. A point in a plane has coordinates (x, y) in S and coordinates (x′,y′)(x′,y′) in S′S′.

(a) Show that, during the transformation of rotation, the coordinates in S′S′ are expressed in terms of the coordinates in S by the following relations:

{ x ′ = x cos φ + y sin φ y ′ = − x sin φ + y cos φ . { x ′ = x cos φ + y sin φ y ′ = − x sin φ + y cos φ .

(b) Show that the distance of point P to the origin is invariant under rotations of the coordinate system. Here, you have to show that

x 2 + y 2 = x ′ 2 + y ′ 2 . x 2 + y 2 = x ′ 2 + y ′ 2 .

(c) Show that the distance between points P and Q is invariant under rotations of the coordinate system. Here, you have to show that

( x P − x Q ) 2 + ( y P − y Q ) 2 = ( x P − x Q ) 2 + ( y P − y Q ) 2 =
( x ' P − x ' Q ) 2 + ( y ' P − y ' Q ) 2 . ( x ' P − x ' Q ) 2 + ( y ' P − y ' Q ) 2 .
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