Period and Velocity of Oscillating Sphere Attached to Spring

In summary: I think you are correct for b) and c). You could also do b) by using the equation of harmonic oscillator ##x=L\sin\omega t## and ##v_{CM}=\dot{x}=\omega L\cos\omega t## and hence setting the max value of $$max( v_{CM})=\omega L=\Omega_{max} R $$
  • #1
Nexus99
103
9
Homework Statement
Work in progress
Relevant Equations
Pure rolling condition
A homogeneous sphere of mass M and radius R is at rest on a rough horizontal plane with coefficient
of static friction μ . A spring of elastic constant k, is connected to the rotation axis of the sphere
illustrated in the figure. The center of mass of the sphere is positioned at rest so that the spring is
lengthened by a stretch equal to L, determine:
a) the period of oscillations of the sphere;
b) the maximum angular velocity of the sphere;
c) the maximum value of L for which the pure rolling speed is maintained.
Cattura.PNG
I tried in this way:
a) if there is an oscillation there is no static friction but dynamic:
Projecting the two cardinal equations along the axis and considering the center of mass of the sphere the point about which the torque is being measured:

-F_{r} - kL = ma
F_{r}R = Ialpha = I a/R = 2/5MRa

F_{r} = 2/5Ma
- 5/7kL = ma k' = 5/7k
-k'L = ma

that is the equation of an harmonic oscillator with ω = √(5/7 k m), that represent the period of oscillations of the sphere, is this solution right?
 
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  • #2
Okpluto said:
if there is an oscillation there is no static friction but dynamic:
That's a strange statement.. did you mean that?
Your equations assume static friction (αR=a).
 
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  • #3
Yes, i realized after that i was wrong but i forgot to delete it
as for the procedure, is it correct?
 
  • #4
Okpluto said:
Yes, i realized after that i was wrong but i forgot to delete it
as for the procedure, is it correct?
This is not quite right:
Okpluto said:
√(5/7 k m)
It is dimensionally wrong. You left out something. Other than that, I think it is ok.
 
  • #5
haruspex said:
This is not quite right:

It is dimensionally wrong. You left out something. Other than that, I think it is ok.
Yes, i forgot a /, it should be:
ω = √(5/7 k/m)
thanks for help
 
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  • #6
Very nice problem ( I like it a a lot really thanks for posting it @Okpluto )
For b) you got to be carefull not to confuse the angular velocity (or angular frequency) of oscillation with the angular velocity of the sphere
for c) you have already derived that for pure rolling we have $$F_r=\frac{2}{5}Ma$$. This becomes max when the acceleration ##a## becomes max and equal to ##\omega^2L##. We have to impose that $$max( F_r)\leq max(static friction)$$.
 
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  • #7
Alternative method of solution: conserve energy. Let $$E = \frac{1}{2}m\dot{x}^2 + \frac{1}{2}I\omega^2 + \frac{1}{2}kx^2 = \frac{1}{2}m\dot{x}^2 + \frac{1}{2}\times \frac{2}{5}mr^2 \times {\left(\frac{\dot{x}}{r}\right)}^2 + \frac{1}{2}kx^2$$ and then set ##\frac{dE}{dt} = 0## since friction does no work during rolling: $$\frac{dE}{dt} = \frac{7}{5}m\dot{x}\ddot{x} + kx\dot{x} = 0 \implies \ddot{x} = -\frac{5k}{7m}x$$
 
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  • #8
I think one should first assume that the system moves without slipping and find the friction force ##F=F(x,h)## where ##x## is the coordinate of the center of disk and ##h## is a constant of the energy integral. Then to consider the condition ##|F(x,h)|\le \mu mg##
 
  • #9
Delta2 said:
Very nice problem ( I like it a a lot really thanks for posting it @Okpluto )
For b) you got to be carefull not to confuse the angular velocity (or angular frequency) of oscillation with the angular velocity of the sphere
for c) you have already derived that for pure rolling we have $$F_r=\frac{2}{5}Ma$$. This becomes max when the acceleration ##a## becomes max and equal to ##\omega^2L##. We have to impose that $$max( F_r)\leq max(static friction)$$.
I solved b and c in this way:

Cattura.PNG


EDIT: $$v_{cm} = \Omega R$$ not small omega
I had to write it in a Latex compiler because for me, in this forum, Latex doesn't always work for
 
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  • #10
Okpluto said:
I solved b and c in this way:

View attachment 263776

EDIT: $$v_{cm} = \Omega R$$ not small omega
I had to write it in a Latex compiler because for me, in this forum, Latex doesn't work (as you can see)
I think you are correct for b) and c). You could also do b) by using the equation of harmonic oscillator ##x=L\sin\omega t## and ##v_{CM}=\dot{x}=\omega L\cos\omega t## and hence setting the max value of $$max( v_{CM})=\omega L=\Omega_{max} R $$
 
  • #11
There seem to be some problems with the rendering of Latex commands, I hope they are going to be fixed soon.
 

Related to Period and Velocity of Oscillating Sphere Attached to Spring

1. What is a sphere attached to a spring?

A sphere attached to a spring is a physical system where a spherical object is connected to a spring, allowing it to oscillate back and forth.

2. How does a sphere attached to a spring work?

The spring provides a restoring force that pulls the sphere back to its equilibrium position when it is displaced. This results in the sphere oscillating around its equilibrium point.

3. What factors affect the motion of a sphere attached to a spring?

The motion of a sphere attached to a spring is affected by the mass of the sphere, the stiffness of the spring, and the amplitude and frequency of the oscillations.

4. What is the equation for the motion of a sphere attached to a spring?

The equation for the motion of a sphere attached to a spring is known as Hooke's law and is given by F = -kx, where F is the force exerted by the spring, k is the spring constant, and x is the displacement from the equilibrium position.

5. What are some real-life applications of a sphere attached to a spring?

Sphere attached to a spring systems are commonly used in pendulum clocks, shock absorbers in vehicles, and as a model for simple harmonic motion in physics experiments.

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