Math 1B
§ 6.4 Work
Overview: Work can be described as the change in energy when a force causes a displacement of an
object. When you carry a refrigerator up a flight of stairs or push a stalled car, you apply a force that
results in the displacement of an object, and work is done. (But if the brake is on and your car doesn’t
move, then you do no “work”!)
If a constant force of magnitude F is applied in the direction of motion of an object and if, as a result,
W = Fd .
the object moves a distance d, then the work W done on the object is
Newton’s Second Law:
F = ma
Units in the SI metric system:
(
€
Units:
Distance
€
mass = kilograms, distance = meters , time = seconds,
F = Newtons N = kg ⋅ m 2
s
€
€
Force
)
€
Work€
€
Example:
a) How much work is done in lifting an 8 kg bucket to a height of 2.5 m? Use the fact that the
acceleration due to gravity (near Earth’s surface) is g ≈ 9.8 m/s2.
b) How much work is done in lifting a 50 lb box 4 ft off the ground?
Note: In part b) we did not have to multiply by g because we were given weight (which is a force) and
not the mass of the object.
Stewart – 7e
1
Calculus enters the picture with variable forces.
Suppose an object moves along the x-axis in the positive direction from x = a to x = b .
€
€
Divide [ a,b] into n subintervals of equal width Δx and let x *i be a sample point of the ith subinterval.
€
€
€
Over a small subinterval the force is almost constant so the work done in moving the object over the
subinterval is ____________________.
We can approximate the total work by
n
Work ≈ ∑ f ( x *i )Δx
i=1
The approximation gets better as the subintervals get smaller (as n → ∞), so
€
n
Work = lim ∑ f ( x *i )Δx
n →∞
i=1
€
But we recognize the right side as a definite integral. Therefore,
€
b
Work =
∫ f ( x )dx
a
Example: Find the work done when a force of
€
10
pounds moves an object along the x-axis from
(1+ x)2
the origin to a point 9 feet from the origin.
Stewart – 7e
2
Hooke’s Law: The force that must be applied to stretch a spring x units beyond its natural length
(equilibrium) is
f ( x ) = kx
where the spring constant k measures the stiffness of the spring.
Example: A spring exerts a force of 5 N when stretched 1 m beyond its natural length of 2.4 m. How
much work is€done to stretch the spring from its natural length to a length of 4.2 m?
Another common work problem arises when the motion is vertical and the force is the gravitational
force. This type of problem becomes interesting when the object being lifted is a body of water, a rope,
or chain. In these situations, different parts of the object are lifted different distances – so integration is
necessary.
Example: A 100 ft length of steel chain weighing 15 lb/ft is used to lift a 1000 lb beam. Find the work
done.
Stewart – 7e
3
Note:
• Water from different levels of the tank is lifted different vertical distances, requiring different
amounts of work.
• Water from the same horizontal plane is lifted the same distance, requiring the same amount of
work.
• A volume V of water has mass 𝜌𝑉, where 𝜌 = 1000 kg/m3 is the mass density of water, OR
• A volume V of water has weight 𝛿𝑉, where 𝛿 = 62.5 lb/ft3 is the weight density of water
Example: The vat shown contains water to a height of 2 m. Find the work required to pump all the
water to the top of the vat.
Stewart – 7e
4