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Problem : Two particles A and B are connected by a rigid rod AB. The rod slides along perpendicular rails as shown here. The velocity of A moving down is 10 m/s. What is the velocity of B when angle θ = 60° ?
Solution : The velocity of B is not an independent velocity. It is tied to the velocity of the particle “A” as two particles are connected through a rigid rod. The relationship between two velocities is governed by the inter-particles separation, which is equal to the length of rod.
The length of the rod, in turn, is linked to the positions of particles “A” and “B” . From figure,
$$x=\sqrt{\left({L}^{2}-{y}^{2}\right)}$$
Differentiatiting, with respect to time :
$$\Rightarrow {v}_{x}=\frac{dx}{dt}=-\frac{2y}{2\sqrt{\left({L}^{2}-{y}^{2}\right)}}X\frac{dy}{dt}=-\frac{y{v}_{y}}{\sqrt{\left({L}^{2}-{y}^{2}\right)}}=-{v}_{y}\mathrm{tan}\theta $$
Considering magnitude only,
$$\Rightarrow {v}_{x}={v}_{y}\mathrm{tan}\theta =10\mathrm{tan}{60}^{0}=10\sqrt{3}\phantom{\rule{1em}{0ex}}\frac{m}{s}$$
Problem : The position vector of a particle is :
$$\begin{array}{l}\mathbf{r}=a\mathrm{cos}\omega t\mathbf{i}+a\mathrm{sin}\omega t\mathbf{j}\end{array}$$
where “a” is a constant. Show that velocity vector is perpendicular to position vector.
Solution : In order to prove as required, we shall use the fact that scalar (dot) product of two perpendicular vectors is zero. Now, we need to find the expression of velocity to evaluate the dot product as intended. We can obtain the same by differentiating the expression of position vector with respect to time as :
$$\begin{array}{l}\mathbf{v}=\frac{d\mathbf{r}}{dt}=-a\omega \mathrm{sin}\omega t\mathbf{i}+a\omega \mathrm{cos}\omega t\mathbf{j}\end{array}$$
To check whether velocity is perpendicular to the position vector, we evalaute the scalar product of r and v , which should be equal to zero.
$$\begin{array}{l}\mathbf{r}\mathbf{.}\mathbf{v}=0\end{array}$$
In this case,
$$\begin{array}{l}\Rightarrow \mathbf{r}\mathbf{.}\mathbf{v}=(a\mathrm{cos}\omega t\mathbf{i}+a\mathrm{sin}\omega t\mathbf{j})\phantom{\rule{2pt}{0ex}}\mathbf{.}\phantom{\rule{2pt}{0ex}}(-a\omega \mathrm{sin}\omega t\mathbf{i}+a\omega \mathrm{cos}\omega t\mathbf{j})\\ \Rightarrow -{a}^{2}\omega \mathrm{sin}\omega t\mathrm{cos}\omega t+{a}^{2}\omega \mathrm{sin}\omega t\mathrm{cos}\omega t=0\end{array}$$
This means that the angle between position vector and velocity are at right angle to each other. Hence, velocity is perpendicular to position vector. It is pertinent to mention here that this result can also be inferred from the plot of motion. An inspection of position vector reveals that it represents uniform circular motion as shown in the figure here.
The position vector is always directed radially, whereas velocity vector is always tangential to the circular path. These two vectors are, therefore, perpendicular to each other.
Problem : Two particles are moving with the same constant speed, but in opposite direction. Under what circumstance will the separation between two remains constant?
Solution : The condition of motion as stated in the question is possible, if particles are at diametrically opposite positions on a circular path. Two particles are always separated by the diameter of the circular path. See the figure below to evaluate the motion and separation between the particles.
Problem : A car of width 2 m is approaching a crossing at a velocity of 8 m/s. A pedestrian at a distance of 4 m wishes to cross the road safely. What should be the minimum speed of pedestrian so that he/she crosses the road safely?
Solution : We draw the figure to illustrate the situation. Here, car travels the linear distance (AB + CD) along the direction in which it moves, by which time the pedestrian travels the linear distance BD. Let pedestrian travels at a speed “v” along BD, which makes an angle “θ” with the direction of car.
We must understand here that there may be number of combination of angle and speed for which pedestrian will be able to safely cross before car reaches. However, we are required to find the minimum speed. This speed should, then, correspond to a particular value of θ.
We can also observe that pedestrian should move obliquely. In doing so he/she gains extra time to cross the road.
From triangle BCD,
$$\begin{array}{l}\mathrm{tan}(90-\theta )=\mathrm{cot}\theta =\frac{\mathrm{CD}}{\mathrm{BC}}=\frac{\mathrm{CD}}{2}\\ \Rightarrow \mathrm{CD}=2\mathrm{cot}\theta \end{array}$$
Also,
$$\begin{array}{l}\mathrm{cos}(90-\theta )=\mathrm{sin}\theta =\frac{\mathrm{BC}}{\mathrm{BD}}=\frac{2}{\mathrm{BD}}\\ \Rightarrow \mathrm{BD}=\frac{2}{\mathrm{sin}\theta}\end{array}$$
According to the condition given in the question, the time taken by car and pedestrian should be equal for the situation outlined above :
$$\begin{array}{l}t=\frac{4+2\mathrm{cot}\theta}{8}=\frac{\frac{2}{\mathrm{sin}\theta}}{v}\end{array}$$
$$\begin{array}{l}v=\frac{8}{2\mathrm{sin}\theta +\mathrm{cos}\theta}\end{array}$$
For minimum value of speed, $\frac{dv}{d\theta}=0$ ,
$$\begin{array}{l}\Rightarrow \frac{dv}{d\theta}=\frac{-8\phantom{\rule{2pt}{0ex}}x\phantom{\rule{2pt}{0ex}}(2\mathrm{cos}\theta -\mathrm{sin}\theta )}{{(2\mathrm{sin}\theta +\mathrm{cos}\theta )}^{2}}=0\\ \Rightarrow (2\mathrm{cos}\theta -\mathrm{sin}\theta )=0\\ \Rightarrow \mathrm{tan}\theta =2\end{array}$$
In order to evaluate the expression of velocity with trigonometric ratios, we take the help of right angle triangle as shown in the figure, which is consistent with the above result.
From the triangle, defining angle “θ”, we have :
$$\begin{array}{l}\mathrm{sin}\theta =2\surd 5\end{array}$$
and
$$\begin{array}{l}\mathrm{cos}\theta =\frac{1}{\surd 5}\end{array}$$
The minimum velocity is :
$$\begin{array}{l}v=\frac{8}{2\phantom{\rule{2pt}{0ex}}x\phantom{\rule{2pt}{0ex}}2\surd 5+\frac{1}{\surd 5}}=\frac{8}{\surd 5}=3.57\phantom{\rule{2pt}{0ex}}m/s\end{array}$$
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