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Grade: 12
        
A collar of mass m slides up the vertical shaft under the action of a force F of constant magnitude but variable direction. If θ = Kt, where K is a constant and if the collar starts from rest with θ = 0. Find the magnitude F of the force which will result in the collar coming to rest as θ reaches π/2. The coefficient of kinetic friction between collar and shaft is μ.
2 years ago

Answers : (3)

Eshan
askIITians Faculty
2095 Points
							Net force acting on the body upwards=Fcos\theta-mg-\mu (Fsin\theta)=ma

\implies \dfrac{dv}{dt}=\dfrac{Fcos\theta-mg-\mu (Fsin\theta)}{m}
=\dfrac{F}{m}(cos(Kt)-\mu sin(Kt))-g
\implies v=\int_0^{\pi/2}(\dfrac{F}{m}(cos(Kt)-\mu sin(Kt))-g)dt
\implies \int_0^0 dv=\int_0^{t_0}(\dfrac{F}{m}(cos(Kt)-\mu sin(Kt))-g)dt
\implies 0=\dfrac{F}{Km}(sin(Kt_0)-\mu +\mu cos(Kt_0))-gt_0

Since\dfrac{\pi}{2}=Kt_0\implies t_0=\dfrac{\pi}{2K}
\implies 0=\dfrac{F}{Km}(1-\mu )-\dfrac{g\pi}{2K}
\implies F=\dfrac{mg\pi}{2(1-\mu)}
one year ago
Gitanjali Rout
184 Points
							
JHVH\implies F=\dfrac{mg\pi}{2(1-\mu)}\implies 0=\dfrac{F}{Km}(1-\mu )-\dfrac{g\pi}{2K}\dfrac{\pi}{2}=Kt_0\implies t_0=\dfrac{\pi}{2K}Since\implies 0=\dfrac{F}{Km}(sin(Kt_0)-\mu +\mu cos(Kt_0))-gt_0\implies \int_0^0 dv=\int_0^{t_0}(\dfrac{F}{m}(cos(Kt)-\mu sin(Kt))-g)dt\implies v=\int_0^{\pi/2}(\dfrac{F}{m}(cos(Kt)-\mu sin(Kt))-g)dt=\dfrac{F}{m}(cos(Kt)-\mu sin(Kt))-g\implies \dfrac{dv}{dt}=\dfrac{Fcos\theta-mg-\mu (Fsin\theta)}{m}Fcos\theta-mg-\mu (Fsin\theta)=maNet force acting on the body upwards=
…............................................................................................................................................................
one year ago
Forum Team
94 Points
							
 
 
Gitanjali, please stop copy pasting answers posted by other users. You will not be rewarded for these points and your account will be blocked due to this activity as it is against forum guidelines. Please check the guidelines here – Forum point Policy – https://www.askiitians.com/forum-point-policy/
 
Thanks
Forum Team
one year ago
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