6.4 arc length. length of a curve in the plane if y=f(x) s a continuous first derivative on [a,b],...
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6.4 Arc Length
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Length of a Curve in the Plane
If y=f(x) s a continuous first derivative on [a,b], the length of the curve from a to b is
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Example:
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6.5 Area of a Surface
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Generated by revolving y=f(x)
about the x-axis
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Generated by revolving y=f(x)
about the y-axis
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Example:
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6.6 Work
Generally
Work=force distance
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Example
A force of 112 N is required to slide a cement block 4 m. How much work is necessary?
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“work” category
6.6 Lifting
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Lifting a bucket of sand Sand weighs 144 lb. Bucket weighs 4 lb. Rope weighs 0.08 lb/ft.
Bucket leaks at a steady rate and the sand is ½ gone when the bucket is lifted 18 ft.
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How Much Work
a) To lift the sand?
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How Much Work
b) To lift the sand and the bucket?
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How Much Work
c) To lift the sand, the bucket and the rope?
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As a bucket is raised 30 ft., water leaks out at a constant rate. Find the work done if the bucket originally contained 24 lb of water and 1/3 leaks out. Weight of empty bucket is 4 lb.
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How Much Work
a) To lift the water?
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How Much Work
a) To lift the water and bucket?
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Bucket Lifted 20 ft Contains 16 lb water ( 2 gal) Leaks at constant rate Empty when reaches top
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How Much Work For water only?
For water and bucket, bucket weighs 5 lb
For water and bucket and rope, rope weighs 0.08 lb/ft.
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“work” category
6.6 Spring
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Hooke’s Lawthe force required to stretch a spring x units beyond is “natural” length is f(x)=k x, k is the constant of the spring.
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k= force length of stretch
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Example
A force of 800 N stretches a spring 0.7 m.
Find the Work done.
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Spring
20 N force Stretches spring 3 m beyond natural
length How Much Work
1. To stretch from natural to 5 m?
2. To stretch one additional m?
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Spring
Natural length is 10 m 8 N force stretches it 1.5 m
How Much Work
1. To stretch from natural to 14 m?
2. To stretch from 11m to 13m?
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Spring
250N force stretches it 30 cm
How Much Work to stretch from 20cm to 50cm?
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“work” category
6.6 Pumping
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Vertical cylindrical tank Diameter = 3 ft Height = 6ft Contains water, 62.5 lb/ ft^3
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H M W to pump the water Of a full tank out over the top of the tank?
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H M W to pump the water
Of a full tank thru a pipe which rises to a height of 4 ft above the top of the tank?
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H M W to pump the water
Of a ½ full tank over the top of the tank?
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H M W to pump the water
Of a ½ full tank out the pipe which rises to a height of 4 ft above the top of the tank?
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Cylindrical Tank 4 m high Radius = 2m Tank buried so top is 1 m below graound
level Full of water, 1000 kg/m^3
HMW to pump full tank to ground level?
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Cylindrical Tank Radius = 5 ft Height = 10 ft Full of water, 62.5 lb/ft^3
HMW to pump to a level 4 ft above ground?
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Above-ground circular pool Diameter = 12 ft Height = 5 ft Depth of water is 4 ft
HMW to empty the pool by pumping the water over the top?
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A open tank has the shape of a right circular cone, full of water.
Diameter of top = 8 ft, height = 6 ft
HMW to empty by pumping over the top?
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A cylindrical gas tank has diameter 3 ft and is 4 ft long. The axis of the tank is horizontal.
HMW to pump the entire contents into a tractor if the opening of the tractor tank is 5 ft above the top of the tank.
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“work” category
6.7 Fluid Force
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is the force of a fluid against a side of a tank.
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Example
Find fluid force on the vertical side of the tank in the diagram, it’s full of water (62.5).
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Example
Glass tank, 3 ft. long Square ends, 1 ft. long Full of water (62.5)
Find force exerted on one end Find force exerted on one side
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Example
See diagram
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Right circular cylindrical tank 90 ft high 90 ft diameter Full of molasses (100 lb/ft^3)
How much force is exerted on bottom 1 ft. band?
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Example
Sheet metal, area = 3 sq. ft. Submerged horizontally in 5 ft. of water
Find the fluid force on the top side.
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Find the fluid force on the vertical side of a tank 3 ft. by 4 ft.
It’s full of water (62.5)
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Find the fluid force on the vertical plate in the diagram..It’s submerged in water (1000 kg/cu.m.
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The vertical side of a form for poured concrete is 10 ft long and 2 ft high.
Determine the force on this part of the form. Concrete weighs 140.7 lb/ft^3.
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Cylindrical gas tank, horizontal axis Tank is ½ full Diameter is 3 ft. Gas weighs 42 lb/ft^3
Find fluid force on an end of the tank.
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6.8 Center of Mass
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6.9 Distance traveled
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6.9 Position shift
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6.9 Volume by slicing
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6.9 Parametric graphing
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6.4-6.9 Test The “key” is to know
how to use the formulae