SATHEE: Kinematics 2 Question 6

Kinematics 2 Question 6

8. The position vector $r$ of particle of mass $m$ is given by the following equation $r (t) = α t^{3} \hat{i} + β t^{2} \hat{j}$ where, $α = \frac{10}{3} m s^{- 3}, β = 5 m s^{- 2}$ and $m = 0.1 k g$ . At $t = 1 s$ , which of the following statement(s) is (are) true about the particle?

(2016 Adv.)

(a) The velocity $v$ is given by $v = (10 \hat{i} + 10 \hat{j}) m s^{- 1}$

(b) The angular momentum $L$ with respect to the origin is given by $L = (5 / 3) \hat{k} N m s$

(c) The force $F$ is given by $F = (\hat{i} + 2 \hat{j}) N$

(d) The torque $τ$ with respect to the origin is given by $τ = - \frac{20}{3} \hat{k} N m$

Show Answer

Answer:

Correct Answer: 8. (a, d)

Solution:

$r = α t^{3} \hat{i} + β t^{2} \hat{j}$

$v = \frac{d r}{d t} = 3 α t^{2} \hat{i} + 2 β t \hat{j}$

$a = \frac{d^{2} r}{d t^{2}} = 6 α \hat{i} + 2 β \hat{j}$ At $t = 1 s$ ,

(a) $v = 3 \times \frac{10}{3} \times 1 \hat{i} + 2 \times 5 \times 1 \hat{j} = (10 \hat{i} + 10 \hat{j}) m / s$

(b) $L = r \times p = \frac{10}{3} \times 1 \hat{i} + 5 \times 1 \hat{j} \times 0.1 (10 \hat{i} + 10 \hat{j})$

$= - \frac{5}{3} \hat{k} N - m s$

(c) $F = m a = m \times 6 \times \frac{10}{3} \times 1 \hat{i} + 2 \times 5 \hat{j} = (2 \hat{i} + \hat{j}) N$

(d) $τ = r \times F = \frac{10}{3} \hat{i} + 5 \hat{j} \times (2 \hat{i} + \hat{j})$

$= + \frac{10}{3} \hat{k} + 10 (- \hat{k}) = \frac{- 20}{3} \hat{k} N - m$

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