A small sphere of mass \( m \) is suspended from the top of a hollow pole through which the cable passes. The cable's free end is pulled inward by the tensile force \( \bar{F} \), whose magnitude is a function of time, such that the length of the cable is a specified function \( l(t) \). The sphere is given an initial velocity that causes it to rotate about the pole, as well as to swing outward from the pole. Determine the equations of motion for the sphere.
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First, we need to define the coordinate system and variables. Let's use polar coordinates, with the origin at the top of the pole. Let the radial distance from the pole to the sphere be r(t), and the angle between the vertical axis and the radial line be θ(t). The Show more…
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Using principles from physics it can be shown that when a cable is hung between two poles, it takes the shape of a curve $y=f(x)$ that satisfies the differential equation $$\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{\rho g}{\mathrm{T}} \sqrt{1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}}$$ where $\rho$ is the linear density of the cable, $g$ is the acceleration due to gravity, and T is the tension in the cable at its lowest point, and the coordinate system is chosen appropriately. Verify that the function $$y=f(x)=\frac{T}{\rho g} \cosh \left(\frac{\rho g x}{T}\right)$$ is a solution of this differential equation.
Differentiation Rules
Hyperbolic Functions
Using principles from physics it can be shown that when a cable is hung between two poles, it takes the shape of a curve $y=f(x)$ that satisfies the differential equation $$\frac{d^{2} y}{d x^{2}}=\frac{\rho g}{T} \sqrt{1+\left(\frac{d y}{d x}\right)^{2}}$$ where $\rho$ is the linear density of the cable, $g$ is the accel- eration due to gravity, $T$ is the tension in the cable at its lowest point, and the coordinate system is chosen appropriately. Verify that the function $$y=f(x)=\frac{T}{\rho g} \cosh \left(\frac{\rho g x}{T}\right)$$ is a solution of this differential equation.
Inverse Functions
Using principles from physics it can be shown that when a cable is hung between two poles, it takes the shape of a curve $ y = f(x) $ that satisfies the differential equation $ \frac {d^2 y}{dx^2} = \frac {pg}{T} \sqrt {1 + (\frac {dy}{dx})^2} $ where $ p $ is the linear density of the cable, $ g $ is the acceleration due to gravity, $ T $ is the tension in the cable at its lowest point, and the coordinates system is chosen appropriately. Verify that the function $ y = f(x) = \frac {T}{pg} \cosh (\frac {pgx}{T}) $ is a solution of this differential equation.
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