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

  • Thread starter Broccoli21
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In summary: OK, so integrating x(t) using the Laplace transform, we get x(t)=Acos(wt)with w=sqrt(k/m) and k=m/r.So, in summary, the sphere rolls without slipping, and the force of friction is equal to the force of the spring minus the force of the mass.
  • #1
Broccoli21
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1
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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  • #2
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?
 
  • #3
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.
 
  • #4
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)...
 
  • #5
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
 
  • #6
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?
 

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

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