Question

4. Two forces P and Q of magnitude 2F and 3F,
respectively, are at an angle with each other.
If the force Q is doubled, then their resultant
also gets doubled. Then, the angle is :
​

Answer 1

Answer:

refer to the attachment.......

angle Is=120°........

Answer 2

The angle is 120°.

Explanation:

Given that,

Magnitude of force P = 2F

Magnitude of force Q = 3 F

We need to calculate the resultant

Using formula of resultant

$R^2=P^2+Q^2+2PQ\cos\theta$

Put the value in to the formula

$R^2=4F^2+9F^2+2\times2F\times3F\cos\theta$

$R^2=4F^2+9F^2+12F^2\cos\theta$...(I)

If the force Q is doubled, then their resultant  also gets doubled.

Using formula of resultant

$R^2=P^2+Q^2+2PQ\cos\theta$

Put the value in to the formula

$(2R)^2=4F^2+(6F)^2+2\times2F\times6F\cos\theta$

$4R^2=4F^2+36F^2+24F^2\cos\theta$...(II)

Divided equation (II) by equation (I)

$\dfrac{4R^2}{R^2}=\dfrac{4F^2+36F^2+24F^2\cos\theta}{4F^2+9F^2+12F^2\cos\theta}$

$4=\dfrac{4+36+24\cos\theta}{4+9+12\cos\theta}$

$52+48\cos\theta=40+24\cos\theta$

$12+24\cos\theta=0$

$\cos\theta=\dfrac{-12}{24}$

$\theta=\cos^{-1}(\dfrac{1}{2})$

$\theta=120^{\circ}$

Hence,  The angle is 120°.

Learn more :

Topic : angle

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