- #1

Amit1011

- 17

- 2

- Homework Statement
- If a particle is almost at the top of a smooth sphere, but not exactly at the very top, and then starts to slide, and the particle also experiences a constant horizontal acceleration(a m/sec^2) in addition to acceleration due to gravity, then what will be the equation between theta (the angle it slides from its initial position) and time, until it flies off the sphere ?

- Relevant Equations
- From Energy conservation: $$\frac{1}{2}mv^{2}=mgR(1-cos\theta )+maRsin\theta $$

From Newton's laws of Motion: $$m\frac{dv}{dt}= mgsin\theta +macos\theta $$

Differentiating eq1 mentioned above, and using eq 2, i got : $$v\frac{dv}{d\theta}=R\frac{dv}{dt}$$

From this, i got:$$ \frac{d\theta}{dt}=\sqrt{(2/R)(g(1-cos\theta )+asin\theta)}$$

After this point, I am not able to understand what substitution or may be other method could be used to solve this differential equation. Its more of a math problem at this point, instead of a physics problem. If possible, let me know how to solve this differential equation. Any help would be welcome.

From this, i got:$$ \frac{d\theta}{dt}=\sqrt{(2/R)(g(1-cos\theta )+asin\theta)}$$

After this point, I am not able to understand what substitution or may be other method could be used to solve this differential equation. Its more of a math problem at this point, instead of a physics problem. If possible, let me know how to solve this differential equation. Any help would be welcome.