Question

Two forces $P$ and $Q$ have a resultant perpendicular to $P$, The angle between the forces is _______.

Answer 1

Step 1: Given data

Two forces $P$ and $Q$.

Resultant perpendicular to $P$.

$$\alpha ={90}^{\circ }$$and $$\tan \alpha =\tan {90}^{\circ }=\mathrm{\infty }$$

Step 2: Formula used

From the figure, we write the formula.

The formula for the angle of the resultant vector of two vectors with respect to one vector is given as: $tan\alpha =\frac{\mathrm{Qsin\theta }}{\mathrm{P}+\mathrm{Qcos\theta }}$

Where,

$P$and$Q$ are the two vectors.

$$\theta$$is the angle between the vectors.

$$\alpha$$is the angle between $P$ the vector and the resultant vector.

Step 3: Replace the values in the formula.

Replacing the values in the formula we get,

$\begin{array}{rcl}\tan \alpha & =& \frac{Q\sin \theta }{P+Q\cos \theta }\\ \tan 90°& =& \frac{Q\sin \theta }{P+Q\cos \theta }\\ \infty & =& \frac{Q\sin \theta }{P+Q\cos \theta }\\ P+Q\cos \theta & =& 0\\ \theta & =& {\cos }^{-1}\left(\frac{-P}{Q}\right)\end{array}$

Therefore the angle between the forces is $\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{-}\mathbf{1}}\left(\frac{-P}{Q}\right)$.