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Q5.1-8PE

Expert-verifiedFound in: Page 188

Book edition
1st Edition

Author(s)
Paul Peter Urone

Pages
1272 pages

ISBN
9781938168000

**Show that the acceleration of any object down a frictionless incline that makes an angle ${\mathbf{\theta}}$**** ****with the horizontal is ${\mathbf{a}}{\mathbf{=}}{\mathbf{gsin}}\mathbf{\left(}\mathbf{\theta}\mathbf{\right)}$ ****. (Note that this acceleration is independent of mass.)**** **

**Acceleration of an object is $\mathrm{a}=\mathrm{gsin}\left(\mathrm{\theta}\right)$.**

**The rate at which velocity changes is referred to as acceleration.**

Due to the component of gravitational force along the inclined object will move downward with certain acceleration as there is a constant external force on the object along the incline.

If friction is absent then the object will move only due to the component of gravitational force.

As block is at rest perpendicular to the inclined surface, therefore net force perpendicular to the inclined is zero so

$\mathrm{N}-\mathrm{mg}\mathrm{sin}\left(90-\mathrm{\theta}\right)=0$

or $\mathrm{N}=\mathrm{mg}\mathrm{cos}\left(\mathrm{\theta}\right)$

Along the inclined there is an external force on the system so applying Newton’s second law along the inclined surface

$\mathrm{Net}\mathrm{force}=\mathrm{mass}\times \mathrm{acceleration}$

$\mathrm{mg}\mathrm{cos}\left(90-\mathrm{\theta}\right)=\mathrm{ma}$

Therefore, $\mathrm{a}=\mathrm{g}\mathrm{sin}\left(\mathrm{\theta}\right)$

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