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Wize University Physics Textbook (Master) > Newton's Laws of Motion: Forces and Dynamics

Spring Force

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Spring Force


  • When springs extend or compress, they "want" to return to their equilibrium position. The force points back to the original position of the spring, and is called a restoring force. (It wants to restore the initial position.)

  • The magnitude of the force is proportional to the distance away from the equilibrium position, Δx=x−xo\Delta x =x-x_o where equilibrium is a point at which the net force on spring is zero.
  • The constant of proportionality is called the spring constant, k.


F⃗s=−kΔx⃗\boxed{ \hspace{3cm} \vec{F}_s = -k\Delta \vec{x} \hspace{3cm}}


  • The negative sign indicates that the force points to the initial position of the spring.
  • This is called Hooke's Law.
  • We can see from this equation that the spring constant must have units of N/m.







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A stretchy rope suspended from the ceiling is 59 cm long when a 70 N weight hangs from it, but is 80 cm long when a 205 N weight hangs from it. Determine the spring constant “k” of the rope.

Solution:


X1=0.59 m
F1=70N=mg
X2=0.8 m
F2=205N
k=?


F⃗=−kΔX⃗\vec{F}=-k\Delta\vec{X}
FH⃗ Hooke′s law=−kΔx⃗ \vec{F_H}\ Hooke's\ law=-k\Delta\vec{x}\
F=mg=FH → Because of equillibrium .F=mg=F_H\ \rightarrow\ Because\ of\ equillibrium\ .

magnitude
F1=kΔx1=k(0.59−xo)F_1=k\Delta x_1=k\left(0.59-x_o\right) k(0.59−xo)=70k\left(0.59-x_o\right)=70
⟹ \Longrightarrow\
F2=kΔx2=k(0.8−xo)F_2=k\Delta x_2=k\left(0.8-x_o\right) k(0.8−xo)=205k\left(0.8-x_o\right)=205

2 equations , 2 unknowns  ↗k\ \nearrow k
↘xo\searrow x_o
0.59k−kxo=700.59k-kx_o=70
0.8k−kxo=205           ⟹   k=643 Nm0.8k-kx_o=205\ \ \ \ \ \ \ \ \ \ \ \Longrightarrow\ \ \ k=643\ \frac{N}{m}

Practice: Hooke's Law

A spring with a spring constant (k) of 300 N/m is compressed 15 cm from its equilibrium position.

What is the magnitude of the restorative force acting on the spring?

Which of the following curves best describes the dependence of the magnitude of the spring force as a function of displacement from equilibrium?


Extra Practice
A 1.2 kg is at rest on a spring at the base of a 35 degree inclined plane. If the spring constant is 105 N/m, and the spring is compressed by 12 cm, how far will it travel up the incline when it is release?
Vertical spring
The springs AA, BBand CChave the same initial length. If kA=kC=0.5kBk_A=k_C=0.5k_B,
a) Find the values of mBm_B, mCm_C.
b) Rank the period of oscillations.
Problems on Momentum Conservation and Collisions
Practice: Conservation of Momentum with Spring
A 15.0-kg block is attached to a very light horizontal spring with spring constant 500 N/m and is resting on a friction-less horizontal table. It is struck by a 3.00-kg stone traveling horizontally at 8.00 m/s to the right. The stone rebounds at 2.00 m/s horizontally to the left. Find the maximum distance that the block will compress the spring after the collision. Enter the answer with 3 significant digits.
Friction and a spring
Practice: Friction and a Spring
A spring with a spring constant k=340 N/m is used to launch a 1.5kg block along the horizontal surface whose coefficient of sliding friction is 0.27. If the spring is compressed 18cm, how far does the block slide?
Hooke's Law and Simple Harmonic Motion
A stretchy rope suspended from the ceiling is 59 cm long when a 70 N weight hangs from it, but is 80 cm long when a 205 N weight hangs from it. Determine the spring constant “k” of the rope.
Momentum and spring
A 15.0-kg block is attached to a very light horizontal spring with spring constant 500 N/m and is resting on a friction-less horizontal table. It is struck by a 3.00-kg stone traveling horizontally at 8.00 m/s to the right. The stone rebounds at 2.00 m/s horizontally to the left. Find the maximum distance that the block will compress the spring after the collision.
Practice: Spring constant
A 300-gram ball starts to roll down 4 meters on a 30 degree incline. (That is, the distance it travels along the slope is 4 meters.) The ball then rolls 2 meters on flat surface and hits a plate, connected to a spring. The spring is compressed 25 cm. If the kinetic coefficient of friction of all surfaces is 0.1, what is the spring constant?
Practice: Friction and a spring
A spring with a spring constant k=340 N/m is used to launch a 1.5kg block along the horizontal surface whose coefficient of sliding friction is 0.27. If the spring is compressed 18cm, how far does the block slide?
Energy conservation and circular motion
A compressed spring is used to propel a 25-gram ball-bearing along a frictionless horizontal track which contains a circular vertical loop of radius 0.30 m in a vertical plane. The spring obeys Hooke's law and is compressed 20 cm by a force of 5.0 N.
a) What is the speed of the ball-bearing at the top of the loop? Treat the ball bearing as a point mass.
b) What is the minimum spring compression required to have the ball-bearing complete a full loop?
Practice: Springs and slopes
A 2.00-kg block is pushed against a spring with negligible mass and force constant k= 400 N/m, compressing it 0.220 m. When the block is released , it moves along a friction-less , horizontal surface and then up a friction-less incline with slope 37.0∘37.0^{\circ}(Fig).
A 2 kg block rests on a spring as shown above, when a 3 kg block is dropped from a height of 3 m above it. The spring constant is 28 N/m. Determine:
The system below initially starts from rest, with a spring constant of 28 N/m.
At t = 0, the 3 kg block is removed. If the 2 kg block is not attached to the spring:
Friction and a spring
A spring with a spring constant k=340 N/m is used to launch a 1.5kg block along the horizontal surface whose coefficient of sliding friction is 0.27. If the spring is compressed 18cm, how far does the block slide?
Problems on Momentum Conservation and Collisions
A 15.0-kg block is attached to a very light horizontal spring with spring constant 500 N/m and is resting on a friction-less horizontal table. It is struck by a 3.00-kg stone traveling horizontally at 8.00 m/s to the right. The stone rebounds at 2.00 m/s horizontally to the left. Find the maximum distance that the block will compress the spring after the collision. Enter the answer with 3 significant digits.
A 1.2 kg is at rest on a spring at the base of a 35 degree inclined plane. If the spring constant is 105 N/m, and the spring is compressed by 12 cm, how far will it travel up the incline when it is release?
Hooke's Law and Simple Harmonic Motion
A stretchy rope suspended from the ceiling is 59 cm long when a 70 N weight hangs from it, but is 80 cm long when a 205 N weight hangs from it. Determine the spring constant “k” of the rope.
Vertical spring
The springs AA, BBand CChave the same initial length. If kA=kC=0.5kBk_A=k_C=0.5k_B,
a) Find the values of mBm_B, mCm_C.
b) Rank the period of oscillations.
This question concerns the great garbage patch in the middle of Pacific ocean. Consider the garbage patch to be a perfect cylinder. At this moment 100 seagulls are sitting on the patch. Every time a wave hits the garbage patch the garbage patch bobs up and down 5 times in 7.2 seconds. When the garbage patch is still the 100 seagulls leave. You know notice that the garbage bobs up and down 5 times in 7 seconds.
The garbage patch weighs 1000 kg. (To make calculations simpler this number is much smaller than the real number.)
a) What is the total mass of the 100 seagulls
A 0.15 kg frame, when suspended from a coil spring, stretches the spring 0.070 m.A 0.200-kg lump of putty is dropped from rest onto the frame from a height of 30.0 cm(Fig). Find the maximum distance the frame moves downward from its initial position.