By Maud King Murphy
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Extra resources for A Beginner's Guide to the Stars
Find the radial and transverse components of the velocity and accelera tion of a particle moving along a plane curve in such a way that r^Ö = 1, given that the equation of the curve is (i) r = 6/(2 - cos 0); (ii) r = 5/(3 + 2 cos 0). 8. Find the velocity and acceleration of a particle moving along the line Γ = a + /7B where a and b are constant vectors, and /? is a scalar function of / such that (i) Ρ = 3ί2; (ü) ρ = At; (iii) ρ = sini; (iv) = / 4- log /. 5. Momentum If at time / a particle of mass m is situated at a point P, whose position vector referred to a fixed origin Ο is r, its linear momentum is defined to be the vector mx.
E. e. e. if 7Γ < 19 < 2π. Thus the vector (sF sin θ)η has the magnitude of the moment of F about O; moreover the vector (sF sin θ)η is positive for 0 < Ö < π and negative for π < θ < 2π, thus it has the sign of the moment of F about O. It is, therefore, convenient to say that the moment of F about Ο is the vector (sF sin θ)η, where a vector in one line impUes a turning effect in a plane perpendicular to that line, in a sense determined by the right-hand screw rule. This suggests that the vector result of multiplying the vector a by the vector b should be the vector (ab sin ö)n,where a, b, η form a right-handed system of vectors and θ is the angle made with the direction of a by the direction of b.
A X a) = —b. (a b) = - c . (b X c), a). X X (Ill) From equations (/) it follows that the value of the scalar triple product is unaltered by changing the order of the letters, provided cyclic order is maintained. From equations (II) it follows that the value of the scalar triple product is unaltered if the dot and cross are interchanged. From equations (III) it follows that if one pair of letters in a scalar triple product is interchanged, thus reversing the cyclic 42 PRODUCTS OF VECTORS order, the sign of the result is changed.