Cylinder attached to a block on an inclined plane, rotational dynamics

In summary: The total mass (Mt) being accelerated = Me + Mb(kg)What?The force due to gravity driving the system = ( M * g * sine θ ) + ( Mb * g * sine θ ) (Newtons)The drag force of the block = Mb * g * cosine Ѳ * µ(Newtons)So :The net force (fn) driving the system :fn = ( ( M * g * sine θ ) + ( Mb * g * sine θ ) ) - ( Mb * g * cosine Ѳ * µ )The acceleration (a) of the system = fn / MtSo :a = ( ( ( M
  • #1
jsrev
4
0

Homework Statement



A cylinder of mass M and radius R, with moment of inertia:
[tex]I_c = \frac{1}{2}MR^2[/tex]
rolls down without slipping through an inclined plane with an angle of θ, while pulling a block of mass Mb with an attached frame (of insignificant mass) which is connected to the axis of the cylinder.
If the cylinder rolls without slipping over the surface and the coefficient of kinetic friction between the block Mb and the surface is μ, then:

a. Deduct an expression for the linear acceleration of the system.

This problem didn't come with a figure, but I made one based on what I think the system looks like.
See figure http://imgur.com/HyiUSLu


Homework Equations



Linear acceleration

The Attempt at a Solution



I've tried a few ways, like calculating the linear acceleration of each object (the cylinder and the block) but there are some things I don't get.
I have tried calculating
[tex]a_{cylinder} = \frac{gsinθ}{k}[/tex]
given that k is a constant that equal 1/2 from the first equation. Then I got the linear acceleration of the block with [tex] a_{block} = gsinθ [/tex] and also considered the friction, but I'm not any close to the answer.

The given solution is:

[tex] a = \frac{2g[(M+M_b)sinθ-μM_bcosθ]}{3M + 2M_b} [/tex]

If anyone can guide me a little bit in this one I would really appreciate it. Thank you.
 
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  • #2
jsrev said:

Homework Statement



A cylinder of mass M and radius R, with moment of inertia:
[tex]I_c = \frac{1}{2}MR^2[/tex]
rolls down without slipping through an inclined plane with an angle of θ, while pulling a block of mass Mb with an attached frame (of insignificant mass) which is connected to the axis of the cylinder.
If the cylinder rolls without slipping over the surface and the coefficient of kinetic friction between the block Mb and the surface is μ, then:

a. Deduct an expression for the linear acceleration of the system.

This problem didn't come with a figure, but I made one based on what I think the system looks like.
See figure http://imgur.com/HyiUSLu


Homework Equations



Linear acceleration

The Attempt at a Solution



I've tried a few ways, like calculating the linear acceleration of each object (the cylinder and the block) but there are some things I don't get.
I have tried calculating
[tex]a_{cylinder} = \frac{gsinθ}{k}[/tex]
given that k is a constant that equal 1/2 from the first equation. Then I got the linear acceleration of the block with [tex] a_{block} = gsinθ [/tex] and also considered the friction, but I'm not any close to the answer.

The given solution is:

[tex] a = \frac{2g[(M+M_b)sinθ-μM_bcosθ]}{3M + 2M_b} [/tex]

If anyone can guide me a little bit in this one I would really appreciate it. Thank you.
Draw a free body diagram for the cylinder, and one for the block.

Here's your image:
attachment.php?attachmentid=70542&stc=1&d=1402529651.png
 

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  • #3
See post 4
dean barry
 
Last edited:
  • #4
Sorry, made a hash of that, here's the revised effort :

Assuming its a homogenous cylinder, then the equivalent mass (Me) of the non slipping cylinder for acceleration calculation purposes = 1.5 * M
(kg)

The total mass (Mt) being accelerated = Me + Mb
(kg)

The force due to gravity driving the system = ( M * g * sine θ ) + ( Mb * g * sine θ )
(Newtons)

The drag force of the block = Mb * g * cosine Ѳ * µ
(Newtons)

So :
The net force (fn) driving the system :
fn = ( ( M * g * sine θ ) + ( Mb * g * sine θ ) ) - ( Mb * g * cosine Ѳ * µ )

The acceleration (a) of the system = fn / Mt
So :
a = ( ( ( M * g * sine θ ) + ( Mb * g * sine θ ) ) - ( Mb * g * cosine Ѳ * µ ) ) / ( Me + Mb )

Comments ?
deanbarry365@yahoo.com
 
  • #5
dean barry said:
Sorry, made a hash of that, here's the revised effort :

Assuming its a homogenous cylinder, then the equivalent mass (Me) of the non slipping cylinder for acceleration calculation purposes = 1.5 * M
(kg)
What?
 

FAQ: Cylinder attached to a block on an inclined plane, rotational dynamics

What is the purpose of a cylinder attached to a block on an inclined plane in rotational dynamics?

The purpose of this setup is to demonstrate the principles of rotational dynamics, including torque, angular acceleration, and moment of inertia. It allows for the study of how forces and motion affect the rotational motion of an object.

How does the angle of the inclined plane affect the motion of the cylinder and block?

The angle of the inclined plane determines the direction and magnitude of the gravitational force acting on the cylinder and block. This, in turn, affects the torque and angular acceleration of the system. A steeper incline will result in a greater torque and faster rotation, while a shallower incline will have less of an effect on the rotational motion.

What is the relationship between the mass of the cylinder and block and their rotational motion?

The mass of the cylinder and block affects the moment of inertia of the system, which is a measure of its resistance to rotational motion. A greater mass will result in a larger moment of inertia, requiring more torque to produce the same angular acceleration. Therefore, the mass of the objects directly influences their rotational motion.

How does the placement of the cylinder on the block affect the rotational dynamics?

The placement of the cylinder on the block affects the distribution of mass and the moment of inertia of the system. The location of the center of mass also plays a role in the torque and angular acceleration. Placing the cylinder closer to the edge of the block will result in a larger moment of inertia and a different distribution of forces, potentially altering the rotational motion.

What factors can cause changes in the rotational motion of the cylinder and block?

The rotational motion of the cylinder and block can be affected by various factors, including the angle of the inclined plane, the mass and placement of the objects, the presence of external forces, and the surface friction between the objects and the inclined plane. Any changes to these factors can alter the torque and angular acceleration, resulting in different rotational motion.

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