Solve First Year Harmonic Motion Problem: Cylinder Rolling

[Xiothus]

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:
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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?
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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!

 

[haruspex]

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.

 

[nasu]

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.

 

[Lnewqban]

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/

 

[Chestermiller]

Let's see your free body diagram on the cylinder.

 

[hutchphd]

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.