Question

Any engineering students help me out in this question ​

Answer 1

Question:-

a uniform cylinder of radius r is spinned about its axis to the angular velocity ω₀ and then placed into a corner.the coefficient of friction between corner walls and cylinderis equal to k. how many turns will the cylinder accomplish before it stops?

$\tt \underbrace{ let's \: understand \: the \: concept}$

In the problem, the rigid body is in translation equilibrium but there is an angular retardation. We first sketch the free body diagram of the cylinder. Obviously the friction forces, acting on the cylinder, are kinetic.

so now, we're comfortable to solve it.

From the condition of translational equilibrium for the cylinder,

$\rm mg = N_1 + kN_2; N_2 = kN_1$

hence, $\rm N_1 = \dfrac{mg}{1+k^2} ; N_2 = k \dfrac{mg}{1+k^2}$

for pure rotation of cylinder about its rotation axis, $\rm N_z = I \beta_z$

➜ $\rm -kN_1 R- kN_2 R = \dfrac{mR^2}{2} \beta_z$

➜ $\rm \beta_z = - \dfrac{2k(1+k)g}{(1+k^2)R}$

$\rm$

now from kinematical equation we have

$\rm \omega^2 = \omega^{2}_{0} + 2 \beta_z \Delta \phi$

$\rm \Delta \phi = \dfrac{\omega_{0}^{2}(1+k^2)R}{4k(1+k)g}$ because ω = 0

hence number of turns

$\rm n = \dfrac{\Delta \phi}{2\pi} = \dfrac{\omega_{0}^{2} (1+k^2)R}{8\pi k(1+k)g}$

Answer 2

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Mathematics is the science that deals with the logic of shape, quantity and arrangement. Math is all around us, in everything we do. It is the building block for everything in our daily lives, including mobile devices, architecture (ancient and modern), art, money, engineering, and even sports.