The 0.8-kg collar slides freely on the fixed circular rod. Calculate the velocity v of the collar...

Question:

The 0.8-kg collar slides freely on the fixed circular rod. Calculate the velocity v of the collar as it hits the stop at B if it is elevated from rest at A by the action of the constant 40-N force in the cord. The cord is guided by the small fixed pulleys.

Figure

Energy

Energy represents the capacity of the body or system to do useful work. The mathematically represents the energy is multiplication force with displacement of body. Its unit is Newton-Meter.

Answer and Explanation:


Given Data:

  • The mass of collar slide is: {eq}{m_c} = 0.8\;{\rm{kg}} {/eq}
  • The constant force act in cord is: {eq}F = 40\;{\rm{N}} {/eq}
  • The vertical distance from point A to B is: {eq}{l_1} = 0.4\;{\rm{m}} {/eq}
  • The distance between point D to C is: {eq}{l_2} = 0.3\;{\rm{m}} {/eq}
  • The initial velocity of collar which is at rest is: {eq}V{ _1} = 0 {/eq}
  • The final velocity of collar is: {eq}V{ _2} {/eq}


The diagram of given problem is

Figure(1)


The expression for distance between point A to C , initial to final condition is:

{eq}{l_d} = \sqrt {l_1^2 + l{}_2^2} {/eq}


Substitute the value in above expression

{eq}\begin{align*} {l_d} &= \sqrt {{{\left( {0.4\;{\rm{m}}} \right)}^2} + {{\left( {0.3\;{\rm{m}}} \right)}^2}} \\ {l_d} &= 0.5\;{\rm{m}} \end{align*} {/eq}


The collar goes from point A to B so it gain potential energy. The expression for potential energy of collar is

{eq}{E_c} = {m_c}{g_o}{l_1} {/eq}


The collar posses initial and final velocity. So it has kinetic energy.

The expression for initial kinetic energy is

{eq}{K_{ei}} = \dfrac{1}{2}{m_c}V_1^2 {/eq}


The expression for final kinetic energy is

{eq}{K_{e2}} = \dfrac{1}{2}{m_c}V_2^2 {/eq}

The expression for change in kinetic energy from initial to final position is

Change in kinetic energy is equal to final kinetic energy minus initial kinetic energy.

{eq}\begin{align*} \Delta {K_e} &= {K_{e2}} - {K_{e1}}\\ \Delta {K_e} &= \dfrac{1}{2}{m_c}V_2^2 - \dfrac{1}{2}{m_c}V_1^2 \end{align*} {/eq}


The force in cord produces so due to this force and with its length it also posses energy.

The expression for energy generate by cord force is

{eq}{E_d} = F{l_d} {/eq}

The expression for energy balance is

{eq}\begin{align*} F{l_d} - {E_c} &= \Delta {K_e}\\ F{l_d} - {m_c}{g_o}{l_1} &= \dfrac{1}{2}{m_c}V_2^2 - \dfrac{1}{2}{m_c}V_1^2 \end{align*} {/eq}


Substitute the value in above expression

{eq}\begin{align*} 40\;{\rm{N}} \times 0.5\;{\rm{m}} - 0.8\;{\rm{kg}} \times 9.81\;{\rm{m/}}{{\rm{s}}^{\rm{2}}} \times 0.4\;{\rm{m}} &= \dfrac{1}{2} \times 0.8\;{\rm{kg}} \times V_2^2 - \dfrac{1}{2} \times 0.8\;{\rm{kg}} \times {\left( 0 \right)^2}\\ {V_2} &= 5.511\;{\rm{m/s}} \end{align*} {/eq}

Thus the velocity of collar when it reaches the point B is {eq}5.511\;{\rm{m/s}} {/eq}


Learn more about this topic:

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What is Energy Conservation? - Definition, Process & Examples

from ICSE Environmental Science: Study Guide & Syllabus

Chapter 1 / Lesson 6
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