By Kleppner D., Kolenkow R.
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Extra resources for An Introduction to Mechanics
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7 Law of sines* Prove the law of sines using the cross product. It should only take a couple of lines. 8 Vector proof of a trigonometric identity Let aˆ and bˆ be unit vectors in the x−y plane making angles θ and φ with the x axis, respectively. Show that aˆ = cos θˆi + sin θˆj, bˆ = cos φˆi + sin φˆj, and using vector algebra prove that cos(θ − φ) = cos θ cos φ + sin θ sin φ. 9 Perpendicular unit vector* ˆ and B = (2ˆi + Find a unit vector perpendicular to A = (ˆi + ˆj − k) ˆj − 3k). 10 Perpendicular unit vectors* ˆ Given vector A = 3ˆi + 4ˆj − 4k, ˆ (a) find a unit vector B that lies in the x−y plane and is perpendicular to A.
U cos θ θ x As the particle moves to the right, θ decreases and rˆ and θˆ change direction. Ordinarily, of course, we try to use coordinates that make the problem as simple as possible; polar coordinates can be used here, but they are not well suited to this problem. 16 Velocity of a Bead on a Spoke A bead moves along the spoke of a wheel at constant speed u meters per second. The wheel rotates with uniform angular velocity θ˙ = ω radians per second about an axis fixed in space. ω u At t = 0 the spoke is along the x axis, and the bead is at the origin.
32 VECTORS AND KINEMATICS v t = t1 t = t2 θˆ1 rˆ 1 b rˆ 2 b θˆ2 θ θ v ˆ The Since r = b = constant, v is purely tangential and v = b α t θ. ˆ and v at a time t1 and at a later time t2 . sketches show rˆ , θ, The particle is located at the position t θ = θ0 + r = b, 0 ˙ = θ0 + 1 αt2 . θdt 2 If the particle is on the x axis at t = 0, then θ0 = 0. The particle’s position vector is r = bˆr, but as the sketches indicate, θ must be given to specify the direction of rˆ . 15 Straight Line Motion in Polar Coordinates Consider a particle moving with constant velocity v = uˆi along the line y = 2.