Question

A particle of mass 3 kg is rotating in a circle of radius 1 m such that the angle rotated by its radius is given by θ = 3 (t + sint). Find the net force acting on the particle when t = π/2

Answer 1

Answer on Question #56925-Physics-Mechanics-Relativity

A particle of mass $3\,\mathrm{kg}$ is rotating in a circle of radius $1\,\mathrm{m}$ such that the angle rotated by its radius is given by $\theta = 3\,(t + \mathrm{sint})$. Find the net force acting on the particle when $t = \pi /2$

Solution

The net force is

$$\vec{F} = m r \hat{\alpha} + m \hat{r} \omega^2.$$ $$\alpha = \frac{d^2}{dt^2} \theta = \frac{d^2}{dt^2} 3(t + \mathrm{sint}) = -3\,\mathrm{sint}.$$ $$\left| \alpha \left(\frac{\pi}{2}\right) \right| = 3 \sin \left(\frac{\pi}{2}\right) = 3.$$ $$\omega^2 = \frac{d}{dt} \theta = \frac{d}{dt} 3(t + \mathrm{sint}) = 3(1 + \cos t).$$ $$\omega^2 \left(\frac{\pi}{2}\right) = \left(3 \left(1 + \cos \frac{\pi}{2}\right)\right)^2 = 3^2 = 9.$$ $$F = 3 \cdot 1 \cdot \sqrt{3^2 + 9^2} = 9\sqrt{10}\,\mathrm{N}.$$

Answer: $9\sqrt{10}\,\mathrm{N}$

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