1. A ball attached to a 3.0m string (pendulum)
A ball attached to a 3.0m string (pendulum) is released from rest at horizontal position. What is the speed of the ball at the lowest point of the swing?
Break it down:
A ball attached to a 3.0m string (pendulum)
is released from rest
What is the speed
of the ball at the lowest point of the swing?
Length of the string L = 3.0m. When the ball reaches the lowest point of the path from the horizontal position, the vertical distance it travels is 3.0m. yi - yf = 3.0m
Vi = 0; initial speed at horizontal position.
at horizontal position.
Let’s place the origin of the coordinating system (0, 0) at this point.  
Vf = ?; final speed when it reaches the lowest point.
Select the lowest point as potential reference level. At this point the ball has zero gravitational potential energy, but maximum kinetic energy.
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2. A ball attached to a 3.0m string (pendulum) is released from rest at horizontal position. What is the speed of the ball at the lowest point of the swing?
Solution:
Draw a diagram +Y
L = 3m
At the top:
Yi = 3.0m Vi = 0
EPi = m g Yi EKi = 0
L = 3m
At the lowest point of swing:
Yf = 0.0m Vf = ?
EPf = m g Yf = EKf = mVf2/2 +X
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3. Wg = EPi – EPf = m*g*yi – m*g*yf  Wg = mg (yi - yf)
A ball attached to a 3.0m string (pendulum) is released from rest at horizontal position. What is the speed of the ball at the lowest point of the swing?
Solution:
Calculate works done by gravitational force (learned from previous problem)
Wg = EPi – EPf = m*g*yi – m*g*yf  Wg = mg (yi - yf)
yi = 3.0m and yf = 0.0
 Wg = m*g*(3 - 0) = 3(m*g) This is the work that can be potentially done by weight when it is at the top.
Apply work-energy theorem (learned in problem 6)
W = EKf - EKi  W = m(Vf2 - Vi2)/2
Vi = 0  W = mVf2/ This is the work that can be done during moving from the top to the lowest point.
Equal these calculated works
3*m*g = mVf2/  3*m*9.8 = mVf2/2  Vf2 = 6*9.8  Vf2 = 58.8
 Vf =7.67m/s
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4. This is the principal of mechanical energy conservation.
A ball attached to a 3.0m string (pendulum) is released from rest at horizontal position. What is the speed of the ball at the lowest point of the swing?
Solution:
In this problem we learned that potential energy can do work. And by work-energy theorem we learned that work is difference in kinetic energy of the object in two different points. Put them together we find that potential energy converts to kinetic energy. In other words;
EPf – EPi = EKi – EKf  EPf + EKf = EPi + EKi
“EP + EK” is called mechanical energy. So an object has the same mechanical energy at each point although potential and kinetic energy in each point are different.
This is the principal of mechanical energy conservation.
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