Multiple Choice
In the context of energy types, is sound energy considered potential or kinetic energy?
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9. Work & Energy
Intro to Energy & Kinetic Energy
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If you know the velocity at the top of a vertical circle, how can you find the velocity at the bottom of the vertical circle?
A
Assume the velocity is the same at the top and bottom due to uniform circular motion.
B
Use conservation of mechanical energy to equate the potential and kinetic energies at the top and bottom.
C
Apply Newton's second law to find the net force at the bottom.
D
Use the centripetal force equation to solve for the velocity at the bottom.
Verified step by step guidance1
Start by understanding the concept of uniform circular motion, where the speed of the object remains constant as it moves along the circular path. This implies that the kinetic energy at the top and bottom of the circle is the same.
Apply the conservation of mechanical energy principle. At the top of the circle, the object has both potential energy due to its height and kinetic energy due to its velocity. At the bottom, the potential energy is zero, and all energy is kinetic. Set up the equation: \( E_{top} = E_{bottom} \), where \( E = KE + PE \).
Express the kinetic energy (KE) and potential energy (PE) using the formulas: \( KE = \frac{1}{2}mv^2 \) and \( PE = mgh \), where \( m \) is mass, \( v \) is velocity, \( g \) is acceleration due to gravity, and \( h \) is height.
Use Newton's second law to find the net force at the bottom of the circle. The net force is the centripetal force required to keep the object moving in a circle, given by \( F_{net} = m \cdot a_c \), where \( a_c = \frac{v^2}{r} \) is the centripetal acceleration and \( r \) is the radius of the circle.
Finally, use the centripetal force equation \( F_c = \frac{mv^2}{r} \) to solve for the velocity at the bottom. Since the velocity is the same at the top and bottom due to uniform circular motion, you can equate the forces and solve for \( v \).
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