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There are 3 steps to solve this one.
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Solution
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Step 1
Given,
θ =the angle of the inclination
θ = 30
o
Free body diagram:
A
60o
10''
10''
60o
B
30o
For the height h3​:
Since the energy of the whole system is conserved, any decrease in
the potential energy of the spheres caused by the drop is converted
into kinetic energy of the spheres.
h 3 = h 1 × sin θ
Where,
h = The drop of the height of sphere 1, from its original position in
the vertical direction (20'')
1
Substitute the values,
h 3 = 20
′′
× sin 30
h 3 = 20 ×
o
1
2
h 3 = 10
Explanation:
For the height h2​:
Since the energy of the whole system is conserved, any
decrease in the potential energy of the spheres caused by the
drop is converted into kinetic energy of the spheres.
h 2 = h 3 + h 3 × sin θ
Where,
h = The drop of the height of sphere 3, from its original
position in the vertical direction (10'')
3
Substitute the values,
h 2 = 10 + 10 × sin 30
h 2 = 10 + 10 ×
o
1
2
h 2 = 15
Step 2
For the kinetic energy of the system:
δT =
1
2
m(v
2
1
+ v
2
2
2
+ v )
3
Where,
v1= The velocity of the sphere 1
v1=v
v2= The velocity of the sphere 2
v2=v/2
v3= The velocity of the sphere 3
v3=0
Substitute the values,
δT =
δT =
δT =
1
2
1
2
5
8
m(v
2
m ×
mv
+ (
5
4
v
v
2
2
)
+ 0)
2
2
Explanation:
For the gravitational potential energy:
δv = −
mg(h 1 +h 2 +h 3 )
12
Where,
h3= The drop of the height of sphere 3, from its original position
in the vertical direction (10'')
h2= The drop of the height of sphere 2, from its original position
in the vertical direction (15'')
h1= The drop of the height of sphere 1, from its original position
in the vertical direction (20'')
Substitute the values,
δv = (−mg)
δv = −
15
4
10+15+20
12
mg
Step 3
For the velocity of drop of the sphere is:
δT + δv = 0
Where,
δT = Kinetic energy of the system
Gravitational potential energy
g = Acceleration due to gravity
δv =
g = 32.2
ft
sec
Explanation:
Substitute the values,
δT + δv = 0
5
8
v
v
v
mv
2
2
2
=
2
−
15
4
15
4
mg = 0
8
mg ×
5
= 6g
= 6 × 32.2
v = √ 6 × 32.2
v = √ 193.2
v = 13.89
ft
sec
Answer
Therefore,
The velocity of drop of the sphere is, v = 13.89
ft
sec
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