A bob of mass m is suspended at a point 'O' by a light string of length 'l' and left to perform vertical motion (circular). Initially applying horizontal velocity Vo at point 'A',... A bob of mass m is suspended at a point 'O' by a light string of length 'l' and left to perform vertical motion (circular). Initially applying horizontal velocity Vo at point 'A', the string becomes slack when the bob reaches point 'D'. What is the ratio of the K.E of the bob at points B and C?
Understand the Problem
The question is asking for the ratio of the kinetic energy of a bob at two different points (B and C) during its vertical circular motion, given that it was initially given a horizontal velocity at point A. The problem involves analyzing the kinetic energy at different points in the motion, implying the use of principles of conservation of energy.
Answer
$$ \frac{KE_B}{KE_C} = \frac{v_B^2}{v_C^2} $$
Answer for screen readers
$$ \frac{KE_B}{KE_C} = \frac{v_B^2}{v_C^2} $$
Steps to Solve
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Define Kinetic Energy Formula The kinetic energy (KE) of an object is given by the formula: $$ KE = \frac{1}{2} mv^2 $$ where ( m ) is the mass of the object and ( v ) is its velocity.
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Determine Velocities at Points B and C To find the ratio of kinetic energies at points B and C, we need to know the velocities at those points. In vertical circular motion, the speed changes due to gravitational potential energy conversion.
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Apply Conservation of Energy At point A, all energy is kinetic. At points B and C, some kinetic energy is converted to potential energy. The total mechanical energy at point A can be expressed as: $$ E_A = KE_A = \frac{1}{2} mv_A^2 $$ At points B and C, their energies include both kinetic and potential parts.
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Express Potential Energy at Different Heights Potential energy (PE) at a height ( h ) is given by: $$ PE = mgh $$ where ( g ) is the acceleration due to gravity. Calculate heights of points B and C from point A.
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Determine Velocities using Energy Conservation The velocities can be derived from conservation of energy: $$ E_A = KE_B + PE_B $$ $$ E_A = KE_C + PE_C $$ From these, if we denote ( h_B ) and ( h_C ) as heights for points B and C respectively, we have: $$ \frac{1}{2} mv_A^2 = \frac{1}{2} mv_B^2 + mgh_B $$ $$ \frac{1}{2} mv_A^2 = \frac{1}{2} mv_C^2 + mgh_C $$
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Calculate the Ratios of Kinetic Energies Reorganize to find velocities ( v_B ) and ( v_C ): $$ v_B^2 = v_A^2 - 2gh_B $$ $$ v_C^2 = v_A^2 - 2gh_C $$ Now, substitute these into the kinetic energy formula: $$ KE_B = \frac{1}{2} m(v_B^2) \quad \text{and} \quad KE_C = \frac{1}{2} m(v_C^2) $$
Finally, find the ratio: $$ \frac{KE_B}{KE_C} = \frac{v_B^2}{v_C^2} $$
$$ \frac{KE_B}{KE_C} = \frac{v_B^2}{v_C^2} $$
More Information
The ratio of kinetic energies at points B and C reflects how speed changes due to gravitational forces during vertical circular motion. Understanding this can help in various applications involving pendulums, roller coasters, and other scenarios where gravitational potential energy is converted to kinetic energy.
Tips
- Neglecting to consider the change in height when applying conservation of energy.
- Confusing potential energy with kinetic energy, leading to incorrect ratios. Remember, total energy is always conserved.
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