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Help needed in circular motion

In circular motion, We know that the angular displacement is represented by $$\theta$$. If a particle moves on circle of radius R, then why is $\hat{\theta}=-sin\theta\hat{i}+cos\theta\hat{j}$

I am a beginner in Kinematics and Circular Motion. All kinds of help accepted.

Note by Gautam Jha
2 years, 7 months ago

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In vector notation the point on a unit circle when at angular position $$\theta$$ is $$cos \theta\hat{i} + sin \theta \hat{j}$$ and its differential w.r.t. $$\theta$$ is given by the expression in consideration.

- 2 years, 7 months ago

That's right.

If you don't yet understand calculus, consider how circular motion is periodic, I.e. if we follow it halfway aaround the circle, its $$x$$ and $$y$$ coordinates are negated. If we draw a line from the origin to the location of the particle, we see that the components are given by sin and cos while the angle is changing at a constant rate $$\theta=\omega t$$. Thus the positions and the velocities must be given by sine and cosine.

If we start motion on the $$x$$ axis, the velocity in the $$x$$ direction must start at zero while the velocity in the $$y$$ direction starts at positive $$\omega$$. Moreover, when we cross the $$y$$ axis, the particle is moving in the negative $$x$$ direction at speed -$$\omega$$, which shows that the $$x$$ velocity is given by the negative sine.

Staff - 2 years, 6 months ago