Question-216788

Question Number 216788 by Tawa11 last updated on 20/Feb/25

Commented by Tawa11 last updated on 20/Feb/25

In Figure DIN11, the mass "m" is located
on an inclined plane of length "L" which
rotates around the vertical axis shown.
The angle between the vertical axis and
the plane is "θ", and the coefficient of static
friction between the plane and the mass is µₛ.
Calculate the range of velocities of the mass
for which it does not slide on the inclined
plane (that is, the minimum and maximum
velocity).

Answered by mr W last updated on 22/Feb/25

Commented by mr W last updated on 22/Feb/25

r=L sin θ for 𝛚_(max) : N=mg sin θ+mω_(max) ^2 r cos θ f=μ_s N=mω_(max) ^2 r sin θ−mg cos θ μ_s (mg sin θ+mω_(max) ^2 r cos θ)=mω_(max) ^2 r sin θ−mg cos θ ω_(max) ^2 L sin θ( sin θ−μ_s cos θ)=g(cos θ+μ_s sin θ) ⇒ω_(max) =(√((g(1+μ_s tan θ))/(L sin θ(tan θ−μ_s )))) for 𝛚_(min) : N=mg sin θ+mω_(min) ^2 r cos θ f=μ_s N=−mω_(max) ^2 r sin θ+mg cos θ μ_s (mg sin θ+mω_(min) ^2 r cos θ)=−mω_(max) ^2 r sin θ+mg cos θ ω_(min) ^2 L sin θ(sin θ+μ_s cos θ)=g(cos θ−μ_s sin θ) ⇒ω_(min) =(√((g(1−μ_s tan θ))/(L sin θ(tan θ+μ_s ))))

$${r}={L}\:\mathrm{sin}\:\theta \\ $$$$\underline{\boldsymbol{{for}}\:\boldsymbol{\omega}_{\boldsymbol{{max}}} :} \\ $$$${N}={mg}\:\mathrm{sin}\:\theta+{m}\omega_{{max}} ^{\mathrm{2}} {r}\:\mathrm{cos}\:\theta \\ $$$${f}=\mu_{{s}} {N}={m}\omega_{{max}} ^{\mathrm{2}} {r}\:\mathrm{sin}\:\theta−{mg}\:\mathrm{cos}\:\theta \\ $$$$\mu_{{s}} \left({mg}\:\mathrm{sin}\:\theta+{m}\omega_{{max}} ^{\mathrm{2}} {r}\:\mathrm{cos}\:\theta\right)={m}\omega_{{max}} ^{\mathrm{2}} {r}\:\mathrm{sin}\:\theta−{mg}\:\mathrm{cos}\:\theta \\ $$$$\omega_{{max}} ^{\mathrm{2}} {L}\:\mathrm{sin}\:\theta\left(\:\mathrm{sin}\:\theta−\mu_{{s}} \mathrm{cos}\:\theta\right)={g}\left(\mathrm{cos}\:\theta+\mu_{{s}} \:\mathrm{sin}\:\theta\right) \\ $$$$\Rightarrow\omega_{{max}} =\sqrt{\frac{{g}\left(\mathrm{1}+\mu_{{s}} \:\mathrm{tan}\:\theta\right)}{{L}\:\mathrm{sin}\:\theta\left(\mathrm{tan}\:\theta−\mu_{{s}} \right)}} \\ $$$$\underline{\boldsymbol{{for}}\:\boldsymbol{\omega}_{\boldsymbol{{min}}} :} \\ $$$${N}={mg}\:\mathrm{sin}\:\theta+{m}\omega_{{min}} ^{\mathrm{2}} {r}\:\mathrm{cos}\:\theta \\ $$$${f}=\mu_{{s}} {N}=−{m}\omega_{{max}} ^{\mathrm{2}} {r}\:\mathrm{sin}\:\theta+{mg}\:\mathrm{cos}\:\theta \\ $$$$\mu_{{s}} \left({mg}\:\mathrm{sin}\:\theta+{m}\omega_{{min}} ^{\mathrm{2}} {r}\:\mathrm{cos}\:\theta\right)=−{m}\omega_{{max}} ^{\mathrm{2}} {r}\:\mathrm{sin}\:\theta+{mg}\:\mathrm{cos}\:\theta \\ $$$$\omega_{{min}} ^{\mathrm{2}} {L}\:\mathrm{sin}\:\theta\left(\mathrm{sin}\:\theta+\mu_{{s}} \:\mathrm{cos}\:\theta\right)={g}\left(\mathrm{cos}\:\theta−\mu_{{s}} \mathrm{sin}\:\theta\right) \\ $$$$\Rightarrow\omega_{{min}} =\sqrt{\frac{{g}\left(\mathrm{1}−\mu_{{s}} \mathrm{tan}\:\theta\right)}{{L}\:\mathrm{sin}\:\theta\left(\mathrm{tan}\:\theta+\mu_{{s}} \right)}} \\ $$

Commented by Tawa11 last updated on 22/Feb/25

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$$$\mathrm{I}\:\mathrm{really}\:\mathrm{appreciate}. \\ $$

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