# Solve First Year Harmonic Motion Problem: Cylinder Rolling

• Xiothus
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
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:
------------------------------------------------------------------------------------------------------------
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!

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.

topsquark
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.

Last edited:
scottdave and topsquark
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.

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/

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

SammyS
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.

## 1. What is harmonic motion?

Harmonic motion refers to the repetitive back-and-forth motion of an object around an equilibrium point, caused by a restoring force that is directly proportional to the displacement from the equilibrium.

## 2. How do you solve a first year harmonic motion problem involving a rolling cylinder?

To solve a first year harmonic motion problem involving a rolling cylinder, you will need to use the equations of motion for a rolling object, which take into account the rotational motion of the cylinder. You will also need to consider the forces acting on the cylinder, such as gravity and friction.

## 3. What are the key variables in a first year harmonic motion problem with a rolling cylinder?

The key variables in a first year harmonic motion problem with a rolling cylinder include the mass of the cylinder, the radius of the cylinder, the amplitude of the motion, the angular frequency, and the initial conditions (such as the initial displacement and velocity).

## 4. How do you determine the period of the motion in a first year harmonic motion problem with a rolling cylinder?

The period of the motion in a first year harmonic motion problem with a rolling cylinder can be determined by using the equation T = 2π/ω, where T is the period and ω is the angular frequency. The angular frequency can be calculated using the equation ω = √(g/R), where g is the acceleration due to gravity and R is the radius of the cylinder.

## 5. Can you use the same equations to solve for the motion of a rolling cylinder on different surfaces?

Yes, the equations used to solve for the motion of a rolling cylinder in a first year harmonic motion problem can be applied to different surfaces. However, the values for variables such as friction and the coefficient of restitution may vary depending on the surface the cylinder is rolling on.

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