Springs
Hooke's 23 (force) law:
F sp s =k s = k s s e
each end exerts this force!
spring constant k
stiffness: demo
Force : stretch ratio
mass m
Force : acceleration ratio
follow up:
WB 10.15 (p. 10-5)
Spring FBDs
draw a FBD for the object on each end
A 20-cm-long spring is attached to a wall. When
pulled horizontally with a force of 100 N, the
spring stretches to a length of 22 cm.
The same spring is now used in a tug-of-war. Two
people pull on the ends, each with a force of 100 N.
What is the value of the spring constant?
How long is the spring while it is being pulled?
The same spring is now suspended from a hook and
a 10.2 kg block is attached to the bottom end.
How long is the stretched spring?
Elastic potential energy
1
2
U sp = k s
2
demo: springs store energy
Usp: another form of potential energy
acts similar to gravitational potential energy
more than one type of potential energy may be present!
K f U sf U gf = K i U si U gi
Ex: A 200 g block on a frictionless surface is pushed against a spring
with spring constant 500 N/m, compressing the spring 2.0 cm. If the
block is released, with what speed is it shot away from the spring?
model:
m = 200 g
k = 500 N/m
Dx = 2.0 cm
system = block+ spring
visualize:
before
after
xe
Ex: A 200 g block on a frictionless surface is pushed against a spring
with spring constant 500 N/m, compressing the spring 2.0 cm. If the
block is released, with what speed is it shot away from the spring?
solve:
1
1
2
m v f U f = K i k x 2
2
2
2
500 N / m kg m / s
k
vf=
x =
0.02 m =1.0 m / s
m
0.2 kg
N
assess:
units?
believable?
reasonable? What happens as we increase...
the compression?
the mass?
the spring constant?
m = 200 g
k = 500 N/m
Dx = 2.0 cm
follow A up: block sliding along a frictionless horizontal
surface with velocity v collides with, and compresses a
spring. The maximum compression is 1.4 cm.
If the block then collides with the spring while
having initial velocity 2v, the spring's maximum
compression will be...
(A) 0.35 cm (B) 0.7 cm
(D) 2.8 cm
(C) 1.4 cm
(E) 5.6 cm
If the spring above is replaced by another with
spring constant twice as large, a block with
velocity v will compress the new spring a
maximum distance...
model
system = block+spring+earth
y = 0 at top of spring arbitrarily
visualize
draw energy bar chart, WB 10-23, page 10-8
after
y2=?
v2=0
y2
y1
A 20-cm-tall spring with spring
constant 5000 N/m is placed
before
vertically on the ground. A 10.2 kg y1=15 cm
block is held 15 cm above the
ye = 20 cm
spring. The block is dropped, hits
v1=0
the spring, and compresses it. What m=10.2 kg
is the height of the spring at the k=5000 N/m
point of maximum compression?
Newton's Laws won't work!
solve
without differential equations
We don't need to find v when it hits the spring!
conservation of energy:
K f U sf U gf = K i U si U gi
1
1
2
2
k y 2 m g y 2= m g y 1 k y 2 m g y 2 m g y 1=0
2
2
Equation is quadratic in y2
Quadratic formula
1
B B 2 4 A C
k y 2 m g y 2 m g y 1 = 0 y 2 =
2
2
2A
solve
1
B B 4 A C
k y 2 m g y 2 m g y 1=0 y 2=
2
2
2A
2
m g m g 2 4 k / 2 m g y 1 m g m g 2 2 k m g y 1
y 2=
=
2 k / 2
k
10.2 kg 9.8 m / s 10.2 kg 9.8 m / s 2 5000 N / m 10.2 kg 9.8 m / s 0.15 m
2
y 2=
2
22
2
2
kg m / s
5000 N / m
N
100 N 400 N
y 2=
=0.1 m , 0.06 m
5000 N / m
Assess:
we expect y2 < 0 y2 = - 10 cm
spring height l = ye+y2 = 20 cm - 10 cm = 10 cm
units? believable?
reasonable? What if we increase m? k? y1?