MAT 21B Lecture Notes - Lecture 12: Riemann Sum, Olive Oil, Specific Weight
6.5 Work and Fluid Forces
Work
When a body moves a distance dalong a straight line by a force Fin the
direction of motion, what is the work done by the force on the body?
Work Done by a Constant Force.
The formula for the work is
W=F·d
where Fis the constant force, and dis the distance traveled.
SI (International System) units
•force: Newton (N);
•distance: meter (m);
•work: joule (J); 1J= 1N·m.
Work Done by a Variable Force along a line
What happens if the magnitude of the force F(x) is a (continuous) function of
the position x?
On each small subinterval [xk−1, xk], the force is approximated by a constant
force F(xk). Thus,
Wk≈F(xk)∆xk
1
Therefore, the total work done from ato bis approximated by the Riemann
sum
W=X
k
Wk≈X
k
F(xk)∆xk
Let the norm of the partition Papproaches zero, we have
W=Zb
a
F(x)dx
That is, work Wdone by the force F(x) is just the definite integral of F(x)!
Example: Spring
Here, the force Fdepends on the position x. What is the formula for F?
Hooke’s law for springs
F=kx
•The constant kis a characteristic of the spring, called the force constant
or the spring constant of the spring.
•xequals to the original nature (uncompressed) length of the spring minus
the new compressed/stretched length.
•x > 0 if it is compressed and x < 0 if it is stretched.
Example: How much work does it take to change a spring from its nature
length Lto a length of L−a?
Example:
A spring has a natural length of
1m. A force of 30Nstretches the
spring to a length of 3m.
2
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MAT 21B Full Course Notes
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Document Summary
W = f d where f is the constant force, and d is the distance traveled. Si (international system) units: force: newton (n), distance: meter (m), work: joule (j); 1j = 1n m. Work done by a variable force along a line. On each small subinterval [xk 1, xk], the force is approximated by a constant force f (xk). Therefore, the total work done from a to b is approximated by the riemann sum. Let the norm of the partition p approaches zero, we have. That is, work w done by the force f (x) is just the de nite integral of f (x)! Here, the force f depends on the position x. A force of 30n stretches the spring to a length of 3m. A 5 lb bucket is lifted from the ground into the air by pulling in. 20 ft of rope at a constant speed.
