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Double
Objective: Work Done by a Variable Force
Work Done by a Spring
Homework: Assignment (1-25)
Do PROBS # (64, 65) Ch. 6, +
Do AP 1986 # 2 (handout)
Date: __________
AP Physics “B”
Mr. Mirro
Date: ________
Work Done by a Variable Force
The work done by a constant force is given by W = (F cos θ) ΔS. Quite often, situations arise in which
the force is NOT constant - but changes with the displacement of the object.
In general, the work done by a variable force in moving an object is equal to the “AREA” under
the graph of F cos θ versus displacement (s) since work = force x distance.
For instance, an archer uses a high-tech instrument called a compound bow. This type of bow consists
of a series of pulleys and strings that produce a force versus displacement.
The work the archer must do in drawing back the string of the compound bow from [0 to 0.5 m] is based
on the variable force that is applied. Consider the graph of F cos θ versus (s) for the compound bow.
The work is equal to the colored area under the curved line in the figure.
For convenience, this area is divided into a
number of small squares, each having an area
of:
Area
A = (9.0 N) x (0.0278 m)
A = 0.25 Joules per square
So, if there are approximately 242 small squares,
the area under the curve is close to:
≈ 242 squares
A = (242 squares) 0.25 Joules
Square
A ≈ 60.5 Joules
AP Physics “B”
Mr. Mirro
Date: ________
Work Done by a Variable Force
Ex 1: The drawing shows the force versus displacement graph for two different bows. These graphs give
the force that an archer must apply to draw the bowstring. [CutnellP6.63]
a) For which bow is more work required to
draw the bow fully from s = 0 to s = 0.5 m ?
b) Estimate the additional work required for the
bow identified in part (a) compared to the
other bow.
Ex 2: A net external force is applied to a 6 kg object that is initially at rest. The net force component
along the displacement of the object varies with the magnitude of the displacement as shown.
[CutnellP6.67]
a) How much work is done by the net force ?
b) What is the speed of the object at s = 20 m ?
AP Physics “B”
Mr. Mirro
Date: __________
Work Done by a Spring
Springs are familiar objects that have many applications, ranging from push-button switches on
electronic components, to automobile suspension systems, to mattresses.
Elastic objects, such as springs, are useful because
they can be either stretched or compressed.
Recall that the elasticity of an object refers to
the object ability to return to its original shape
after “deformation.”
Experiments done by Robert Hooke reveal that for relatively small displacements, the force Fapplied
required to stretch or compress a spring is directly proportional to the displacement Δx.
That is, Hooke’s Law for the restoring force on an ideal spring can be expressed as:
Fapplied = - k Δx
where (k) is the spring constant and
(Δx) is the displacement from equilibrium
The spring constant (k) is a measure of the relative stiffness of the spring.
‰
The higher the spring constant, the “stiffer” the spring.
‰
The stiffer the spring, the more “force” required to deform the spring.
Note: The minus sign in the formula for the force indicates that the restoring force always points in a
direction opposite the displacement of the spring.
We have seen that when an object is allow to fall, such as the
hammer of a pile driver, it can do WORK !
Work in the form of the potential energy due to gravity.
PEg = mg ΔH
A spring also has potential energy when the spring is stretched or
compressed which we refer to as elastic potential energy.
DefN: Elastic Potential Energy
The elastic potential energy PEs is the energy that a spring has by virtue of being temporarily
deformed. For an ideal spring that has a spring constant (k) and is stretched or compressed by
an amount (Δx) relative to its equilibrium (unstrained) position, the elastic potential energy is:
PEs = ½ k (Δx)2
Consider the following system, where the total mechanical energy of this system is:
‰
Entirely elastic potential energy at A
‰
Partly elastic potential energy and partly kinetic at B
‰
Entirely kinetic at C
If we consider the total mechanical energy and apply the
conservation of energy principle to the spring system shown,
we can develop the following expression:
Efinal
=
KE + PEg + PEs =
Einitial
KE + PEg + PEs
½ mv2 + mghf + ½ kx2 = ½ mv2 + mghf + ½ kx2
AP Physics “B”
Mr. Mirro
Date: ________
Work Done by a Spring
Ex 1: The diagram shows an object of mass m = 0.20 kg that is vibrating on a horizontal frictionless
table. The spring has a spring constant k = 545 N/m. The spring is stretched initially to
x0 = 4.50 cm and then is released from rest (v0 = 0). [Cutnell10.7]
a. Write an expression for the final velocity vf in terms of the
spring constant (k), mass (m) and the displacement (x0) and (xf).
b. Determine the final translational speed vf of the object when the final displacement of the
spring is at:
(i) xf = 2.25 cm
and
(ii) xf = 0 cm
Ex 2: A 0.40 kg ball is attached to a vertical spring as shown. The spring constant of the spring is
28 N/m. The ball, supported initially so that the spring is neither stretched nor compressed,
is released from rest. In the absence of air resistance, how far does the ball fall before being
brought to a momentary stop by the spring ? [Cutnell10.9]
Ex 3: One end of a spring whose spring constant k = 50 N/m, is attached to a solid wall while the other end
reaches to the edge of a horizontal, frictionless tabletop, which is a distance h = 1m above the floor. A
block of mass M = .5 kg is placed against the end of the spring and pushed toward the wall until the spring
has been compressed a distance ΔX = 0.2m, as shown below. [AP1986.2sim]
The block is released, follows the trajectory shown,
striking the floor a horizontal distance D = 0.9 m from
the edge of the table. If the air resistance is negligible,
Find the following:
a.
The time elapsed from the instant the block leaves
the table to the instant it strikes the floor.
b.
The horizontal component of the velocity of the block just before it hits the floor.
c.
The work done on the block by the spring.