A Different Angle

Classical Mechanics Level 2

$\theta= 4{ t }^{ 2 }$

The angular position of a particle is determined by the above equation, and it has a radius of rotation of 1 meter. Find the time taken by the particle to complete 2 rotations.

\sqrt{\frac{\pi}{2}}

none of these

\sqrt{\pi}

2 solutions

Swagat Panda
Jun 23, 2015

Since the given expression is quadratic hence we have to use differential calculus to find the acceleration on the particle.

$\dfrac { d\theta }{ dt } =8t\Rightarrow \alpha =\frac { d(8t) }{ dt } =8m{ s }^{ -2 }\\ \\$

Now using second equation of rotational kinematics we get $4\pi ={ \omega }_{ 0 }^{ }t+\frac { 1 }{ 2 } \alpha { t }^{ 2 }\\ \\$

The initial velocity ' ${ \omega }_{ 0 }^{ }$ ' of the particle is 0 as $\dfrac { d\theta }{ dt } =8t$ is 0 at $t=\SI{2}{\second}$

Solving for 't' we get $4\pi =4{ t }^{ 2 }\Rightarrow t=\sqrt { \pi } \\ \\$

Nice way to apply this math witchcraft, but you needn't have calculated the distance that would equal double the perimeter ; instead, you can take angle=4 pi for 2 rotations whatever the diameter is and you'll get the same result.

Taha Ahmed - 5 years, 11 months ago

Yes. You are right! That's why I love posting problems because people like you can give me much shorter and effective solutions. Thanks.

Swagat Panda - 5 years, 11 months ago

John Taylor
Jun 24, 2015

I actually said something way too easy now that i think about it. Since 2 rotations of a particle would give an angle of 4 pi, 4 pi = 4t^2. then you divide by 4 and take the square root of both sides to get t= sqrt pi.

A Different Angle

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