Solve First Year Harmonic Motion Problem: Cylinder Rolling

In summary: The solution for the sliding case is: ω = √(k/M). However, for the rolling case, there is an additional rotational energy term that must be taken into account. The correct value of ω for the rolling case is ω = √(k/M + (1/2)(M)(R^2)/I), where I is the moment of inertia of the cylinder. In summary, the conversation discussed a practice problem related to harmonic motion and work + energy. The question involved a cylinder rolling without slipping and attached to a spring, and the goal was to find the value of ω consistent with constant total energy. The conversation also provided tips for properly conceptualizing and approaching problems in a first year university course
  • #1
Xiothus
1
0
Homework Statement
Hey guys, I'm a first year university student and I'm having trouble with this practice problem. I don't even really know where to start and thought this would be a good place to post for some help. The question is relating to harmonic motion and work + energy and goes as follows:
------------------------------------------------------------------------------------------------------------
Question:
A cylinder of mass M and radius R has an axle through its center. The
cylinder rolls without slipping back and forth along the x-axis. It has a
moment of inertia of (1/2)(M)(R^2). The axle is attached to a spring of spring constant k.
The origin is the place at which the mass experiences no force.
The cylinder is observed to undergo harmonic motion in the form x =A cos (ωt + φ).
What value of ω is consistent with the total energy being
constant?
------------------------------------------------------------------------------------------------------------
Relevant Equations
Circular motion equations + work and energy equations + angular momentum equations: up to first year university.
x = A cos (ωt + φ)
v = -Aωsin (ωt + φ)
a = -A(ω^2)cos (ωt + φ)
Thank you guys for taking the time to read this - I'm decently struggling with first year and need some tips on how to properly conceptualize problems and learn what the right approach is on certain problems.
Have a wonderful day, again thank you for checking this post out!
 
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  • #2
Xiothus said:
Homework Statement: Hey guys, I'm a first year university student and I'm having trouble with this practice problem. I don't even really know where to start and thought this would be a good place to post for some help. The question is relating to harmonic motion and work + energy and goes as follows:
------------------------------------------------------------------------------------------------------------
Question:
A cylinder of mass M and radius R has an axle through its center. The
cylinder rolls without slipping back and forth along the x-axis. It has a
moment of inertia of (1/2)(M)(R^2). The axle is attached to a spring of spring constant k.
The origin is the place at which the mass experiences no force.
The cylinder is observed to undergo harmonic motion in the form x =A cos (ωt + φ).
What value of ω is consistent with the total energy being
constant?
------------------------------------------------------------------------------------------------------------
Relevant Equations: Circular motion equations + work and energy equations + angular momentum equations: up to first year university.
x = A cos (ωt + φ)
v = -Aωsin (ωt + φ)
a = -A(ω^2)cos (ωt + φ)

Thank you guys for taking the time to read this - I'm decently struggling with first year and need some tips on how to properly conceptualize problems and learn what the right approach is on certain problems.
Have a wonderful day, again thank you for checking this post out!
It doesn’t say where the other end of the spring is attached. Is there a diagram?
If not, assume it is somewhere on the same horizontal line as the axle.
Draw a free body diagram and write the acceleration equations (linear and rotational) when at displacement x.
 
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  • #3
You already know the equation for displacement versus time. I would assume that by "x" they mean the position of the center of mass. From this you can find the velocity of the COM and then write the total energy of the system: kinetic energy for translation of the COM, rotation around the COM and potential elastic.
 
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  • #4
Xiothus said:
Homework Statement: Hey guys, I'm a first year university student and I'm having trouble with this practice problem. I don't even really know where to start and thought this would be a good place to post for some help. The question is relating to harmonic motion and work + energy and goes as follows:
...
Thank you guys for taking the time to read this - I'm decently struggling with first year and need some tips on how to properly conceptualize problems and learn what the right approach is on certain problems.
Have a wonderful day, again thank you for checking this post out!
Welcome, @Xiothus!

Try to understand the subject as deeply as your limited time allows you to do.
Then, learn the common approach to specific problems by studying resolved ones if available, and by trying yourself or with some help, including this site.
Don't let difficult problems make you feel insecure or to scare you, most are solvable following basic steps.

Please, see:
https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/15-1-simple-harmonic-motion/

https://courses.lumenlearning.com/s...hapter/15-2-energy-in-simple-harmonic-motion/

:cool:
 
  • #5
Let's see your free body diagram on the cylinder.
 
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  • #6
Can you answer the same question if the system was a sliding (no dissipation "frictionless") block on a spring? Write it down
The method here is the same but there is some rotional energy to consider.
 

FAQ: Solve First Year Harmonic Motion Problem: Cylinder Rolling

What is the basic setup of a first-year harmonic motion problem involving a cylinder rolling?

The basic setup typically involves a solid cylinder rolling without slipping on a surface, often attached to a spring. The problem usually requires analyzing the system's motion, determining the equilibrium position, and finding the angular frequency of the oscillations.

How do you determine the equilibrium position for a rolling cylinder attached to a spring?

The equilibrium position is where the net force on the system is zero. For a cylinder attached to a spring, this is where the spring force (kx, where k is the spring constant and x is the displacement) balances the gravitational force component acting along the direction of motion. For horizontal setups, the equilibrium position is typically where the spring is neither stretched nor compressed.

What equations of motion are used to describe the harmonic motion of the rolling cylinder?

The equations of motion are derived from Newton's second law and the rotational analog. For a cylinder of mass m and radius R, rolling without slipping, the key equations are F = ma for linear motion and τ = Iα for rotational motion, where τ is the torque, I is the moment of inertia (I = 0.5mR² for a solid cylinder), and α is the angular acceleration. The no-slip condition relates linear acceleration a to angular acceleration α by a = Rα.

How do you find the angular frequency of the oscillations for the rolling cylinder?

The angular frequency ω of the oscillations can be found by solving the differential equation of motion for the system. For a cylinder rolling without slipping, the combined linear and rotational motion leads to an effective mass of (3/2)m in the harmonic oscillator equation. The angular frequency is given by ω = sqrt(k / (3/2)m), simplifying to ω = sqrt(2k / 3m).

What assumptions are typically made in solving first-year harmonic motion problems involving a rolling cylinder?

Common assumptions include: the cylinder rolls without slipping, meaning the point of contact with the surface has zero relative velocity; the surface is frictionless except for the rolling condition; the spring follows Hooke's law (F = -kx); air resistance and other damping forces are negligible; and the motion is small enough to be considered simple harmonic.

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