How Do You Solve a Rolling Sphere Attached to a Spring Problem?

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Homework Help Overview

The problem involves a solid sphere attached to a spring, where the sphere rolls without slipping. The task is to find the position of the sphere's center of mass as a function of time, the force of friction, and the path of a point on the sphere in relation to time. The context includes rotational dynamics and the effects of friction in a dynamic system.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss setting up differential equations and the role of moment of inertia in the context of rolling motion. There are attempts to relate translational and rotational dynamics, and questions arise about how to incorporate friction and its effects on the system.

Discussion Status

There are various lines of reasoning being explored, including the derivation of equations for both rolling and non-rolling cases. Some participants suggest starting with simpler cases to build understanding, while others express confusion about the role of friction and how to express forces in the context of rolling motion.

Contextual Notes

Participants note the complexity of incorporating friction into the equations of motion and question whether friction is constant throughout the motion. There is also mention of homework constraints and the specific requirements for solving the problem.

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I am pretty lost as to how to solve this problem :confused:

Homework Statement


A solid sphere of mass m (homogeneously distributed throughout) and radius r is attached to a spring of spring constant k whose other end is attached to a sturdy wall. (assume that the spring is attached at the sphere's center, but still allows it to roll)
(a) Assuming that the sphere rolls without slipping. Find the position of
the sphere’s center of mass x (as a function of time) using only rotational dynamics. At time zero, the ball is stationary at distance A away from its equilibrium position.
(b) Find the force of friction as a function of position.
(c) If P is the point on the sphere touching the ground at t=0, find the path of P in terms of t

Homework Equations


Newton's 2nd:
F=ma
torque=I(alpha)
I=(2/5)mr^2
F_s=force of spring=-kx (x=0 is equilibrium)
F_k = friction = umg
w=angular velocity
when rolling without slipping, Rv=w, R(alpha)=a
total force on ball = F_s + F_k

The Attempt at a Solution


I really have no idea as to how to solve this. I could set up a differential equation, but for Newton's law, the direction of the frictional force F_k changes direction, and I don't know how to factor that in. I can do part (c) easily after that.
Thanks in advance for the help!
 
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Broccoli21 said:
I am pretty lost as to how to solve this problem :confused:

Homework Statement


A solid sphere of mass m (homogeneously distributed throughout) and radius r is attached to a spring of spring constant k whose other end is attached to a sturdy wall. (assume that the spring is attached at the sphere's center, but still allows it to roll)
(a) Assuming that the sphere rolls without slipping. Find the position of
the sphere’s center of mass x (as a function of time) using only rotational dynamics. At time zero, the ball is stationary at distance A away from its equilibrium position.
(b) Find the force of friction as a function of position.
(c) If P is the point on the sphere touching the ground at t=0, find the path of P in terms of t

Homework Equations


Newton's 2nd:
F=ma
torque=I(alpha)
I=(2/5)mr^2
F_s=force of spring=-kx (x=0 is equilibrium)
F_k = friction = umg
w=angular velocity
when rolling without slipping, Rv=w, R(alpha)=a
total force on ball = F_s + F_k

The Attempt at a Solution


I really have no idea as to how to solve this. I could set up a differential equation, but for Newton's law, the direction of the frictional force F_k changes direction, and I don't know how to factor that in. I can do part (c) easily after that.
Thanks in advance for the help!

Welcome to the PF.

Perhaps it's best to start slow, and build up to the solution the way the problem asks for it.

For a simpler situation, a non-rolling mass on a frictionless horizontal plane, how do you derive the SHM position as a function of time, x(t)?

Then add in the ball rolling on a surface instead of a mass sliding on a frictionless surface. What-all changes? Where does the moment of inertia I for the ball come in? Can you derive an equation for x(t) using both translational and rotational dynamics?

Then, can you see how you can incorporate the translational portion into a full rotational solution?
 
Solving for it when non-rolling is super easy, you simply get x(t)=Acos(wt) with w=sqrt(k/m).
I just have no idea as to how to factor in moment of inertia or friction.
I have tried using the conditions for rolling without slipping, but that still doesn't help with incorporating I...Maybe I could express F_k as a function of theta (the angle the ball has rotated through), then x=r(theta),F_s=-kr(theta), which really doesn't help me...:frown:
I am also not sure as to how friction plays a role. I understand how normal things roll without friction, and can solve for it, but I am lost as to how to do it when the object is an oscillating ball.
 
Broccoli21 said:
Solving for it when non-rolling is super easy, you simply get x(t)=Acos(wt) with w=sqrt(k/m).
I just have no idea as to how to factor in moment of inertia or friction.
I have tried using the conditions for rolling without slipping, but that still doesn't help with incorporating I...Maybe I could express F_k as a function of theta (the angle the ball has rotated through), then x=r(theta),F_s=-kr(theta), which really doesn't help me...:frown:
I am also not sure as to how friction plays a role. I understand how normal things roll without friction, and can solve for it, but I am lost as to how to do it when the object is an oscillating ball.

But how do you derive the frictionless case from the original differential equations? Can you write those out?

The friction just comes in for the ball to provide the torque to generate the angular acceleration that is associated with the final linear acceleration that you integrate to get your x(t)...
 
well the diffE for the simple case is:
x''=(-k/m)x
question: Is friction constant, because F_k=umg right? Why are they asking me to solve for F_k per time? Thats confusing, but back to the first part:
I could write Sum(F)=ma=F_k+F_s, so
a=x''=F_k/m+F_s/m=ug-kx/m
Is this the right equation? I really doubt it. It has nothing to do with rolling...
I really am missing something ugh
Thanks for the help so far though :DDDD
 
well torque is rF_k=I(alpha), and ar=alpha, so
alpha=rF_k/I=ar
a=F_k/I and because of Newton's second law,
a=(F_k-F_s)/m
so solving for F_k we get
mF_k-IF_k=-F_s
F_k=-F_s/(m-I)
is this valid?
 

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