Download Spring problem and Hooke`s Law

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0430 860 810 Nick Zhang
Lecture 4 Spring problem and conservation of mechanical energy
Hooke's Law
The restoring force exerted by the spring is directly proportional to its displacement.
The restoring force acts in a direction that would restore the spring to its natural length.
Mathematically, F = −kx
where
k is the spring constant which is the measure of how stiff the spring is. The SI unit is N/m.
x is the displacement of the end of the spring form its natural position
F is the restoring force
The negative sign indicates that the restoring force is always in the opposite direction to the displacement.
natural length of the spring
x
the spring is compressed and x is the compression
x
the spring is extended and x is the extension
NOTE
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Hooke's Law applies to springs within certain limits. If the spring is compressed or extended so much
that it is permanently deformed, Hooke's Law no longer applies.
The restoring force F and the force that is compressing or extending the spring are a pair of
action-reaction forces and they are equal in magnitude and opposite in direction.
x is the measure of compression or extension which is the change in length of the spring.
The F − x graph produces a straight line with gradient of k .
restoring force (F)
F
extension or compression ( x )
x
The strain potential energy stored in a spring can be found by calculating the area under F - x graph:
Esp =
1
1
1
Fx = kx ⋅ x = kx 2
2
2
2
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F (N )
0430 860 810 Nick Zhang
stiff spring
weak spring
The gradient of the F − x graph
is the spring constant k which
indicates the stiffness of the
material
x ( m)
Conservation of mechanical energy
Recall that mechanical energy is comprised of kinetic energy, strain potential energy and gravitational
potential energy.
If the gravitational force and the restoring force are the only forces doing work on an object, the kinetic
energy and the potential energy of the object interconvert while the total mechanical energy of the system
(comprising the object and the earth or spring) remains constant.
Mathematically,
Eki + E pi = Ekf + E pf if the weight and the restoring force are the only forces doing work on the object
Example
A model rocket of mass 0.20 kg is launched by means of a spring, as shown in figure below. The spring is
initially compressed by 20 cm, and the rocket leaves the spring as it reaches its natural length. The
force-compression graph of the spring is also shown in figure below.
How much energy is stored in the spring when it is compressed?
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0430 860 810 Nick Zhang
What is the speed of the rocket as it leaves the spring?
What is the maximum height, above the spring, reached by the rocket? The air resistance on the way up
could be ignored.
Example
In a lab class at school, Nick is given a spring with a spring constant of 15 N/m and natural length of 0.30 m.
He hangs it vertically, and attaches a mass to it, so that the new length of the spring is 0.50 m.
Assuming the spring has no mass, what was the value of the mass he attached?
Nick pulls the mass down a further distance of 0.10 m.
By how much has the potential energy stored in the spring changed?
He now releases the mass, so that the mass-spring system oscillates. Ignore air resistance.
Which one of the curves (A-D) below could best represent the variation of the total mechanical energy of the
mass-spring system as a function of position?
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0430 860 810 Nick Zhang
Example
Figure on the left shows an ideal spring with a 2.0 kg mass attached. The spring-mass system is held so that
the spring is not extended. The mass is gently lowered and the spring stretches until, in figure on the right,
the spring-mass system is at rest. The spring has extended by 0.40 m.
What is the value of the spring constant, k , of the spring?
What is the difference in the magnitude of the total mechanical energy of the mass-spring system between
the two figures?
Example
Part of a roller coaster ride at an amusement park is shown in figure above. The car with people in it has a
total mass of 800 kg. The cars starts from rest at point A, a vertical height of 20 m above point B.
Ignore the effects of friction.
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0430 860 810 Nick Zhang
What is the speed of the car at point B?
Example
A
h
B
The ball starts from rest at point A, a height of h metres above point B. The ball comes to rest momentarily
at point B due to friction. If the ball starts at point B with an initial speed of 6.0 m/s down the slope, it will
come to a stop at point A. What is the value of h ?
Elastic or inelastic collision
Recall that if two objects collide on a horizontal and smooth surface, then the total momentum of the system
comprising the two objects is conserved. But there are differences between two types of collisions.
1. Elastic collision - the total kinetic energy after the collision is the same as it was before the collision.
BUT during the collision, some kinetic energy of the two colliding objects is transformed into the strain
potential energy due to the temporary deformation of the objects. Since the collision is elastic and the
interaction between the two objects can be considered as restoring forces, therefore, the total energy of the
system is conserved before, during and after the collision.
Total Ek
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Total energy (kinetic and strain potential)
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2. Inelastic collision - even though total momentum is still conserved, the total kinetic energy of the
colliding objects after the collision is less than it was before the collision due to permanent deformation of
the objects. A proportion of the initial kinetic energy is lost in forms of heat and sound.
Total Ek
Total energy (kinetic and strain potential)
Example
Ranjiv, who has a mass of 80 kg, is running with a speed of 4.0 m/s as he steps onto a stationary trolley of
mass 50 kg as shown in figure below. Ranjiv holds on to the trolley. Ranjiv and the trolley then move
forward together in the same direction.
Is this collision between Ranjiv and the trolley elastic or inelastic?
Nick and Sarah are studying collision by sliding blocks together on a frictionless table. Nick slides a block
of mass 2.0 kg with a speed of 3.0 m/s that collides with a block of mass 1.0 kg, which was at rest. After the
collision the 1.0 kg block has a speed of 4.0 m/s. The situations before and after are shown in figure below.
Determine whether this is an elastic or inelastic collision.
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