Question

A particle of mass m is projected with a speed u at an angle θ with the horizontal. Find the torque of the weight of the particle about the point of projection when the particle is at the highest point.?

Answer 1

Hey there !

Solution :

Refer to the attachment for the reference purpose.

The horizontal distance from starting point to the end point is the Range. The horizontal distance marked as W is the Force.

So According to the formula,

Torque = Range × Force

The Magnitude Of Torque = Force × Range SinФ

=> Magnitude of Torque = F.r.SinФ

We know Force = Mass × Gravity

=> F = mg

Now the Curly Bracket marked is Half the Range. Hence it can be said as 

Range / 2 which can be expressed as r / 2.

So substituting in the formula we get,

Torque = mg * r / 2

r can be written as :

r = [ u² . sin 2 Ф / g ]

We know that,

Sin 2Ф = 2 Sin Ф . Cos Ф 

So substituting in the formula we get,

Torque = mg / 2 * [ u² 2 sin Ф. cos Ф / g ]

2 gets cancelled and g also gets cancelled. Hence we get,

Torque = m [ u² sin Ф . cos Ф ]

=> Torque = Mass * Speed² Sin Ф . Cos Ф

In terms of formula we can write it as ,

T = M U² Sin Ф. CosФ

Hope my answer helped !

Answer 2

Hey friend, Harish here.

Here is your answer.

Given:

Mass of the Object = m

Speed = U

Angle made with horizontal = θ

To Find:

The torque of the weight of the particle.

Solution:

$\mathrm{We \ know\ that\ :}\\ \\ \mathrm{Torque(\tau ) = Force(W)\times Distance(d)_{\perp \ to\ action}} \\ \\ \mathrm{Here:}\\ \\ \to \mathrm{Force\ is\ same\ as \ the\ weight}\\ \\ \to \mathrm{Distance\ perpendicular\ to}\ \mathrm{the\ action\ is\ half\ of\ the\ range(R).} \\ \\ \to \tau = W \times \frac{R}{2} \\ \\ \mathrm{We\ know\ that\ :}\\ \\ \to \mathrm{Weight(W)= mass(m) \times acceleration\ due\ to\ gravity(g)=m\times g} \\ \\ \to \mathrm{R= \frac{u^2\sin(2\theta)}{g}} \\ \\ \mathrm{Therefore\ :}$ 

$\to \tau = W \times \frac{R}{2} \\ \\ \to \mathrm{\tau = mg\times \frac{u^2\sin(2\theta)}{2g}}} \\ \\ \to \boxed{\mathrm{\tau = m. \frac{u^2\sin(2\theta)}{2}}}$  
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Refer to the image for good understanding. 

Hope my answer is helpful to you.