A small sphere B of mass m is attached to an inextensible cord of length 2a, which passes around the fixed peg A and is attached to a fixed support at O. The sphere is held close to the support at O and released with no initial velocity. It drops freely to Point C, where the cord becomes taut, and

Question

A small sphere B of mass m is attached to an inextensible cord of length 2a, which passes around the fixed peg A and is attached to a fixed support at O. The sphere is held close to the support at O and released with no initial velocity. It drops freely to Point C, where the cord becomes taut, and swings in a vertical plane, first about A and then about O. Determine the vertical distance from line OD to the highest Point C′′ that the sphere will reach.

Accepted Answer

Velocity at Point C (before the cord is taut).

Conservation of energy from B to C :

[latex]\begin{aligned}T_{B} & =0 \V_{B} & =m g(2)\left\lgroup\frac{\sqrt{2}}{2} ight group a=m g a \sqrt{2} \T_{C} & =\frac{1}{2} m u_{C}^{2} \quad V_{C}=0 \T_{B}+V_{B} & =T_{C}+V_{C} \0+m g a \sqrt{2} & =\frac{1}{2} m u_{C}^{2}+0 \ u_{C} & =\sqrt{2 \sqrt{2}} g a\end{aligned}[/latex]

Velocity at C (after the cord becomes taut).

Linear momentum perpendicular to the cord is conserved:

[latex]\begin{aligned}& heta=45^{\circ} \& \overset{+}{ warrow} -m u_{C} \sin heta=m u_{C}^{\prime} \& u_{C}^{\prime}=(\sqrt{2 \sqrt{2}})\left\lgroup\frac{\sqrt{2}}{2} ight group \sqrt{g a} \& u_{C}^{\prime}=\sqrt{\sqrt{2} g a}=2^{\frac{1}{4}} \sqrt{g a}\end{aligned}[/latex]

Note: The weight of the sphere is a non-impulsive force.

Velocity at C :

[latex]\begin{aligned}\underline{C ext { to } C^{\prime}} ext { (conservation of energy): } \quad T_{C} & =\frac{1}{2} m\left( u_{C}^{\prime} ight)^{2} & V_{C} & =0 \T_{C^{\prime}} & =\frac{1}{2} m\left( u_{C^{\prime}}^{\prime} ight)^{2} & V_{C^{\prime}} & =0\end{aligned}[/latex]

Datum:

[latex]\begin{aligned}T_{C}+V_{C} & =T_{C^{\prime}}+V_{C^{\prime}} \\frac{1}{2} m\left( u_{C}^{\prime} ight)^{2}+0 & =\frac{1}{2} m\left( u_{C}^{\prime} ight)^{2}+0 \ u_{C}^{\prime} & = u_{C^{\prime}}^{\prime}\end{aligned}[/latex]

[latex]\underline{C^{\prime}}[/latex] to [latex]\underline{C^{\prime \prime}}[/latex] (conservation of energy)

[latex]\begin{aligned}& T_{C^{\prime}}=\frac{1}{2} m\left( u_{C^{\prime}}^{\prime} ight)^{2} \& T_{C^{\prime}}=\frac{1}{2} m\left(2^{1 / 4} \sqrt{g a} ight)^{2} \& T_{C^{\prime}}=\frac{\sqrt{2}}{2} m g a\end{aligned}[/latex]

Datum:

[latex]\begin{aligned}T_{C^{\prime}}+V_{C^{\prime}} & =T_{C^{\prime \prime}}+V_{C^{\prime \prime}} \V_{C^{\prime}} & =0 \T_{C^{\prime \prime}} & =0 \V_{C^{\prime \prime}} & =m g h \\frac{\sqrt{2}}{2} m g a+0 & =0+m g h \h & =\frac{\sqrt{2}}{2} a\end{aligned}[/latex]

[latex]h=0.707 \mathrm{\ a}\blacktriangleleft[/latex]

Question 13.196: A small sphere B of mass m is attached to an inextensible co...

This Question has been Answered.

Already have an account?

Login

Related Answered Questions