How Is Acceleration Calculated for a Hollow Cylinder Lawn Roller?

[John O' Meara]

A lawn roller in the form of a hollow cylinder of mass M is pulled horizontally with a constant force F applied by a handle attached to the axle. If it rolls without slipping, find the acceleration and the frictional force.
Let R1 be the radius of the hollow and R2 the outer radius, and (alpha) the angular acceleration.
Sum Fy= n - M*g
Sum Fx= F - f.
Where n = normal reaction and f = friction force. Applying Sum F = M*a, we get n=M*g ...as ay=0.
F - f = M*ax ...the equation for the translational motion of the center of mass. And where ax and ay are the accelerations in the x and y directions resprctively.
And the equation of rotational motion about the axis through the center of mass is:
f*R2 = I*(alpha)=.5*M*(R1^2+R2^2)
f*R2 = .5*M*(R1^2+R2^2)*ax/R2: let a=ax
f = .5*M*(R1^2+R2^2)*a/R2^2 ...(ii)
F-f = M*a...(i) After adding i and ii
a=F/(M+.5*M*(R1^2+R2^2)/R2^2.
The question is, is this correct or not? And where is it wrong?

 

[Doc Al]

Looks OK to me.

 

[kyle8921]

I would say that your approach and equations seem to be correct based on the given information. However, there are a few things that could be clarified or improved upon in your response.

Firstly, it would be helpful to provide some context or background information about rotational inertia and its importance in this problem. This would help the reader understand why the equations and approach you have used are relevant and necessary.

Secondly, it would be beneficial to define your variables and explain their significance in the problem. For example, it would be helpful to define R1 and R2 as the inner and outer radii of the cylinder, respectively, and explain why they are important in determining the frictional force.

Additionally, you could clarify the meaning of the equations you have used, such as the equation for rotational motion about the center of mass. This would help the reader understand the logic behind your approach and how the equations are related to each other.

Finally, it would be good to mention any assumptions that were made in solving this problem, such as assuming that the roller is a rigid body and that there is no external torque acting on it.

Overall, your response seems to be correct, but providing more context, defining variables, and clarifying equations and assumptions would make it more thorough and clear.