Conservation of Energy for an Oscillating Steel Ball The motion of a steel ball in a hemispherical bowl of radius h shown in Fig. 2–51 is to be analyzed. The ball is initially held at the highest location at point A, and then it is released. Obtain relations for the conservation of energy of the

Question

Conservation of Energy for an Oscillating Steel Ball The motion of a steel ball in a hemispherical bowl of radius h shown in Fig. 2–51 is to be analyzed. The ball is initially held at the highest location at point A, and then it is released. Obtain relations for the conservation of energy of the ball for the cases of frictionless and actual motions.

Accepted Answer

A steel ball is released in a bowl. Relations for the energy balance are to be obtained.
Assumptions The motion is frictionless, and thus friction between the ball, the bowl, and the air is negligible.
Analysis When the ball is released, it accelerates under the influence of gravity, reaches a maximum velocity (and minimum elevation) at point B at the bottom of the bowl, and moves up toward point C on the opposite side.
In the ideal case of frictionless motion, the ball will oscillate between points
A and C. The actual motion involves the conversion of the kinetic and potential energies of the ball to each other, together with overcoming resistance to motion due to friction (doing frictional work). The general energy balance for any system undergoing any process is

[latex]\underbrace{E_{ ext {in }}-E_{ ext {out }}}_{\begin{array}{c} ext { Net energy transfer } \ ext { by heat, work, and mass }\end{array}}=\underbrace{\Delta E_{ ext {system }}}_{\begin{array}{c} ext { Change in internal, kinetic, } \ ext { potential, etc., energies } \end{array}} [/latex]

Then the energy balance for the ball for a process from point 1 to point 2 becomes

[latex] -w_{ ext {friction }}=\left( ke _{2}+ pe _{2} ight)-\left( ke _{1}+ pe _{1} ight) [/latex]

or

[latex] \frac{V_{1}^{2}}{2}+g z_{1}=\frac{V_{2}^{2}}{2}+g z_{2}+w_{ ext {friction }} [/latex]

since there is no energy transfer by heat or mass and no change in the internal energy of the ball (the heat generated by frictional heating is dissipated to the surrounding air). The frictional work term [latex] W_{ ext {friction }} [/latex] is often expressed as [latex] e_{ ext {loss }} [/latex] to represent the loss (conversion) of mechanical energy into thermal energy. For the idealized case of frictionless motion, the last relation reduces to

[latex] \frac{V_{1}^{2}}{2}+g z_{1}=\frac{V_{2}^{2}}{2}+g z_{2} \quad ext { or } \quad \frac{V^{2}}{2}+g z=C= ext { constant } [/latex]

where the value of the constant is C = gh. That is, when the frictional effects are negligible, the sum of the kinetic and potential energies of the ball remains constant.
Discussion This is certainly a more intuitive and convenient form of the conservation of energy equation for this and other similar processes such as the swinging motion of the pendulum of a wall clock.

Question 2.14: Conservation of Energy for an Oscillating Steel Ball The mot...

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