Question

A particle is moving along a circular path of radius $5cm$, at a speed of $v=5t^2$ , where $v$ is in $cm/s$ and $t$ is in seconds. The magnitudes of tangential acceleration and total acceleration (in $mm/s^2)$ of the particle at time $t=2s$ are

• A. $200\text{and}825$
• B. $20\text{and}800$
• C. $200\text{and}1000$
• D. $100\text{and}825$

Answer 1

The correct option is A $200\text{and}825$

Given, radius of circular path $r=5cm$
and $v=5t^{2}cm/s$
To find total acceleration :
Total acceleration : $\overrightarrow{a}=\overrightarrow{a_c}+\overrightarrow{a_t}$
$|\overrightarrow{a}|=\sqrt{a_c^2+a_t^2}$
Now,
Centripetal acceleration, $|\overrightarrow{a_c}|=\frac{v^2}{r}=\frac{25t^4}{5}cm/s^2=\frac{25\times (2{)}^4}{5}=5\times 16=80cm/s^2$
Tangential acceleration, $|\overrightarrow{a_t}|=\frac{dv}{dt}=\frac{d}{dt}\left(5t^2\right)=10t=10\times 2=20cm/s^2$
Hence, $|\overrightarrow{a}|=\sqrt{(80{)}^2+(20{)}^2}=\sqrt{6400+400}=\sqrt{6800}\approx 82.5cm/s^2=825mm/s^2$