Question

The position of an object having mass \(0.1\) kg as a function of time \(t\) is given as} \[ \vec r = (10t^2\hat{i}+5t^3\hat{j})\ \text{m}. \] At \(t=1\) s, which of the following statements are correct?} A.} Linear momentum \( \vec p = (2\hat{i}+1.5\hat{j})\) kg m/s. B.} Force acting on the object \( \vec F = (2\hat{i}+3\hat{j})\) N. C.} Angular momentum about origin \( \vec L = 15\hat{k}\) J s. D.} Torque about origin \( \vec \tau = 20\hat{k}\) N m. Choose the correct answer.

• A. A, B and C only
• B. B, C and D only
• C. A, C and D only
• D. A, B and D only

Answer 1

The Correct Option is C

Step 1:Find velocity} \[ \vec v=\frac{d\vec r}{dt} \] \[ \vec v=(20t\hat{i}+15t^2\hat{j}) \] At \(t=1\): \[ \vec v=(20,15) \]
Step 2:Linear momentum} \[ \vec p=m\vec v \] \[ =0.1(20\hat{i}+15\hat{j}) \] \[ =(2\hat{i}+1.5\hat{j}) \] Thus statement A is correct. 
Step 3:Force \[ \vec a=\frac{d\vec v}{dt} \] \[ \vec a=(20\hat{i}+30t\hat{j}) \] At \(t=1\): \[ \vec a=(20,30) \] \[ \vec F=m\vec a \] \[ =0.1(20\hat{i}+30\hat{j}) \] \[ =(2\hat{i}+3\hat{j}) \] Thus B is correct.  
Step 4:Angular momentum} \[ \vec L=\vec r \times \vec p \] At \(t=1\): \[ \vec r=(10,5) \] \[ \vec p=(2,1.5) \] \[ \vec L= \begin{vmatrix} \hat{i}&\hat{j}&\hat{k}\\ 10&5&0\\ 2&1.5&0 \end{vmatrix} \] \[ =(15)\hat{k} \] Thus C is correct. 
Step 5:Torque} \[ \vec \tau=\vec r\times\vec F \] \[ \vec F=(2,3) \] \[ \vec \tau= \begin{vmatrix} \hat{i}&\hat{j}&\hat{k}\\ 10&5&0\\ 2&3&0 \end{vmatrix} \] \[ =20\hat{k} \] Thus D is correct. Hence the correct statements are \[ \boxed{A,\ C,\ D} \]