A small particle of mass m is projected at an angle \Theta with the  x-axis with an initial velocity \upsilon _{0} in the  x-y plane as shown in the figure. At a time t< \frac{\upsilon _{0}\sin \Theta }{g}, the angular momentum of the particle is

where \hat{i},\hat{j}\, and\, \hat{k} are unit vectors along x,y and z-axis respectively.

  • Option 1)

    \frac{1}{2}mg\upsilon _{0}t^{2}\cos \Theta \hat{i}

  • Option 2)

    -mg\upsilon _{0}t^{2}\cos \Theta \hat{j}

  • Option 3)

    mg\upsilon _{0}t\cos \Theta \hat{k}

  • Option 4)

    -\frac{1}{2}mg\, \upsilon _{0}t^{2}\cos \Theta \hat{k}

     

 

Answers (1)

As we discussed in

1st equation or velocity time equation -

 

V=u+at

V = Final velocity

u = Initial velocity

A = acceleration

T = time

-

 

 \vec{v}_x = \left ({v_{0}} \right \cos \Theta)\hat{i}

\vec{v}_y = \left ( v_{0} \right\sin \Theta -gt )\hat{j}

\vec{v} ={v}_x\hat{i}\:+{v}_{y}\hat{j}

\vec{v} = v_{o}\cos \Theta \hat{i} + \left (v _{0} sin \Theta \right - gt )\hat{j}

\vec{r} = v_{o} \cos \Theta \hat{i} + \left (v_{o} \sin \Theta t- {\tfrac{1}{2}} gt^2 \right )\hat{j}

Angular momentum

= \vec{L} = m\left ( \vec{r} \times \right\vec{v} )

L = m (v_0cos\Theta \hat{i}+(v_0sin\Theta t-\frac{1}{2}gt^2)\vec{j})\times (v_0cos\Theta \hat{i}+(v_0sin\Theta-gt))\hat{j}

L = m [v^2_0cos\Theta sin\Theta t - v_0gt^2cos\Theta]\hat{k}+(v^2_0sin\Theta cos\Theta t - \frac{1}{2}gt^2v_0cos\Theta (-\hat{k})

L =m (v^2_0sin\Theta cos\Theta t \hat{k} - v_0gt^2 cos\Theta (\hat{k}) - v^2_0sin\Theta cos\Theta t \hat{k} +\frac{1}{2}mgv_0t^2cos\Theta \hat{k}

L =m[- \frac{1}{2}v_0gt^2cos\Theta \hat{k}]

L =\frac{1}{2}mgv_0t^2cos\Theta \hat{k}


Option 1)

\frac{1}{2}mg\upsilon _{0}t^{2}\cos \Theta \hat{i}

Incorrect

Option 2)

-mg\upsilon _{0}t^{2}\cos \Theta \hat{j}

Incorrect

Option 3)

mg\upsilon _{0}t\cos \Theta \hat{k}

Incorrect

Option 4)

-\frac{1}{2}mg\, \upsilon _{0}t^{2}\cos \Theta \hat{k}

 

Correct

Most Viewed Questions

Preparation Products

Knockout JEE Main April 2021 (One Month)

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 14000/- ₹ 4999/-
Buy Now
Knockout JEE Main May 2021

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 22999/- ₹ 9999/-
Buy Now
Test Series JEE Main May 2021

Unlimited Chapter Wise Tests, Unlimited Subject Wise Tests, Unlimited Full Mock Tests, Get Personalized Performance Analysis Report,.

₹ 6999/- ₹ 2999/-
Buy Now
Knockout JEE Main May 2022

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Live Classes, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 34999/- ₹ 24999/-
Buy Now
JEE Main Rank Booster 2021

Booster and Kadha Video Lectures, Unlimited Full Mock Test, Adaptive Time Table, 24x7 Doubt Chat Support,.

₹ 13999/- ₹ 6999/-
Buy Now
Boost your Preparation for JEE Main 2021 with Personlized Coaching
 
Exams
Articles
Questions