Question 16: Work, Energy and Power & Circular Motion

Question

An object of mass 1000 g experiences a time dependent force $\vec{F}=\left(2 t \hat{i}+3 t^2 \hat{j}\right) N$. The power generated by the force at time $t$ is:

Accepted Answer

Given Information:
Force: $\vec{F} = (2t \hat{i} + 3t^2 \hat{j}) \, 	ext{N}$
Mass of the object, $m = 1000 \, 	ext{g} = 1 \, 	ext{kg}$
Determine Acceleration:
Using Newton's second law, $\overrightarrow{F} = m \overrightarrow{a}$.
Thus, the acceleration, $\overrightarrow{a} = 2t \hat{i} + 3t^2 \hat{j}$.
Velocity Calculation:
To find velocity, integrate acceleration with respect to time:
$    \frac{d\overrightarrow{v}}{dt} = 2t \hat{i} + 3t^2 \hat{j}    $
Integrate to find $\overrightarrow{v}$:
$    \overrightarrow{v} = \int (2t \hat{i} + 3t^2 \hat{j}) \, dt = t^2 \hat{i} + t^3 \hat{j} + \vec{C}    $
(Assuming initial velocity is zero, $\vec{C} = 0$, hence $\overrightarrow{v} = t^2 \hat{i} + t^3 \hat{j}$).
Calculate Power:
Power is defined as the dot product of force and velocity vectors:
$    P = \overrightarrow{F} \cdot \overrightarrow{v} = (2t \hat{i} + 3t^2 \hat{j}) \cdot (t^2 \hat{i} + t^3 \hat{j})    $
Computing the dot product gives:
$    P = (2t \cdot t^2) + (3t^2 \cdot t^3) = 2t^3 + 3t^5    $
The power generated at time $t$ is $P = (2t^3 + 3t^5) \, 	ext{W}$.

Answer

$\left(2 t^2+18 t^3\right) W$

Answer

$\left(3 t^3+5 t^5\right) w$

Answer

$\left(2 t^3+3 t^5\right) w$

Answer

$\left(2 t^2+3 t^3\right) w$

Work, Energy and Power & Circular Motion

An object of mass 1000 g experiences a time dependent force \vec{F}=\left(2 t \hat{i}+3 t^2 \hat{j}\right) N. The power generated by the force at time t is:

Choose the correct answer: