Motion in two dimensions, in a plane can be studied by expressing position, velocity and acceleration as vectors in Cartesian co-ordinates where are unit vector along x and y directions, respectively and Ax and Ay are corresponding components of (Fig). Motion can also be studied by expressing vectors in circular polar co-ordinates as  A=A_rr + ${\mathit{A}}_{\mathit{\theta }}\mathit{\theta }$  where ? =r/r=cos $\mathit{\theta }\mathit{i}+\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathit{j}$  and $\mathit{\theta }=-\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathit{i}+\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\mathit{j}$ are unit vectors along direction in which 'r' and 'q ' are increasing. 

(a) Express in terms of q.

(b) Show that both q are unit vectors and are perpendicular to each other.

(c) Show that d(r)/dt , where w =dq/dt q and dq/dt = -wr

(d) For a particle moving along a spiral given by r=aqr , where a = 1 (unit), find dimensions of 'a'.

(e) Find velocity and acceleration in polar vector representation for particle moving along spiral described in (d) above.

A ball is projected from ground at angle q with the horizontal. After t = 1 s, it is moving at 45° with the horizontal and after t = 2 second, it is moving horizontally. What is speed of projection of ball? [g = 10 m s^–2]

 

An electron in a research apparatus follows a circle path. On the electron’s first circuit of the apparatus, its speed is v₀ and the radius of its circular path is R. Each time it makes on circuit, it passes a short region where it receives a “kick” and gains an additional speed of $\frac{v_0}{100}.$  The electron follows a circular path such that the magnitude of its acceleration is always the same. What is the radius of the circular path after the electron has received 10 kicks?

Two particles start moving from the same point in two different directions. The first particle moves along the positive x-axis with a constant acceleration of 3 m/s². The second particle moves in a straight line making an angle of 60° with the positive x-axis with a constant speed of 24 m/s. Find the time after which the relative velocity of the particles is minimum.

A particle is moving in a circle of radius 1m with angular velocity (ω) given as a function of angular displacement ( θ ) by the relation ω = θ² + 2θ. The total acceleration of the particle when θ = 1rad is:
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A flying disc of radius r is moving with constant speed v? while rotating anticlockwise with angular speed v?/r along a curve PQ as shown below. Radius of curvature of curve at the instant is 4r. The magnitude of acceleration of point A at the instant is (3v₀²)/(nr). Find n.