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I recently performed an experiment for a class which involved measuring the projectile motion of a ball after it had rolled down a ramp and exited on a significantly smaller ramp. When interpreting the measurements taken (which include the time taken for the ball to leave the ramp, diameter of the ball along with a video of the flight path), I made the mistake of applying the motion of an object sliding down a plane instead of that of a ball rolling down a plane- which I have not covered and only have limited mathematical understanding of.

I have found a number of formulas online including Angular acceleration, Moment of Inertia and Rolling friction but I am either not using them correctly, I am missing something or I am completely overthinking the process. The ultimate goal of this is to find the velocity of the ball as it leaves the ramp, am I just overthinking this?

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I would assume that the friction does not dissipate energy but only causes the ball to roll without slipping. Then, the potential energy $$E_p = m g h$$ will be converted to kinetic energy $$E_{pot} = \frac{1}{2} m v^2$$ and to angular momentum. The energy due to the angular momentum can be calculated via the moment of inertia, which, for a solid ball is $$I= \frac{2}{5} m r^2$$ with corresponding energy $$E_{rot}= \frac{1}{2} I \omega^2.$$ With the condition for the rolling and not slipping $$\omega=\frac{v}{r}$$ you should be able to solve the problem $$E_{pot}=E_{rot}+E_{kin}.$$ Good luck!

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FORCE/TORQUE METHOD:

$mg\sin(\theta)-f=ma$

$fR=I\alpha$

$a_c=R\alpha$

$\implies a_c=\dfrac{g\sin(\theta)}{(1+k^2 /R^2)}$ and $f=\dfrac{mg\sin(\theta)}{(1+R^2/k^2)} $

$v_{final}^2=0+2(a_c)(\dfrac{h}{\sin(\theta)})$

Solve to get $v_{final}$

MECHANICAL ENERGY CONSERVATION:

$0+mgh=\dfrac 12 mv_{final} ^2 +\dfrac 12 I_{c}\omega_{final}^2$

$v_{final}=R\omega_{final}$

$W_{friction}=0$ since point of contact of the spherical body with inlined plane is always instantaneously at rest.

Solve to get $v_{final}$

WORK-ENERGY THEOREM SEPARATELY IN TRANSLATIONAL MOTION AND ROTATIONAL MOTION:

$-f(\dfrac{h}{\sin(\theta)})+mgh=\dfrac{1}{2}mv_{final}^2$

$(fr)(\dfrac{h/\sin(\theta)}{r})=\dfrac{I_{c}w_{final}^2}{2}$ [Toque * Angular displacement = Final Rotational KE]

Solve to get $v_{final}$

ANGULAR IMPULSE MOMENTUM THEOREM METHOD:

$L=L_{translational}+L_{rotational}$

By applying the impulse-momentum theorem at any random point at time $t$ and $t+dt$:

$( \dfrac{I_{c}}{r_{final}} + mr )(v)(- \hat k) + mgr\sin(\theta)dt (-\hat k)=( \dfrac{I_{c}}{r_{final}} + mr )(v + dv)(- \hat k)$

Find $\dfrac{dv}{dt}$ from above equation which equals $a_{c}$ and then use it to find final velocity at bottom as shown previously.

You can use linear impulse momentum theorem also like: $mv +mg\sin(\theta) dt -fdt = m(v+dv)$ to solve the problem.Just reuse the value of $f$ we obtained in first method!

Hope this helps!

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