Question

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.)

Short answer

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

Step-by-step solution

Definition of acceleration

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

Determining acceleration of any object down a frictionless incline

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

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

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

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

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

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

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