Mechanics (rolling motion of ball on cart)

In summary, the conversation discusses a problem involving a homogeneous disk resting on a cart and being pushed by a force. The minimum coefficient of static friction required to prevent slipping, the acceleration of the cart, the angular acceleration of the disk, and the linear acceleration of the disk are all being determined using various equations and attempts. The correct solution involves finding the pseudo force in a non-inertia frame and ultimately results in a linear acceleration of 0 for the disk.
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Homework Statement



A homogeneous disk A of a radius R is resting on a cart B (on wheels, assume no work against friction) when a force F is applied to the cart. Assuming the slip does not slip, determine:
(a) minimum coefficient of static friction required to prevent slipping
(b) acceleration of the cart
(c) angular acceleration of the disk
(d) linear acceleration of the disk

Mass of A and B are also known. Let mass of A be Ma and mass of B be Mb.

Actual numerical values are as follow:
Ma=20kg
Ma=30kg
R=0.6m
F=60N

Homework Equations



F=ma=umg (where u is coefficient of friction, static)
I=mr2/2 (where I is moment of inertia of a disk)
FR=I*alpha (where alpha is angular acceleration)
No slip: v=rw, a=R*alpha (where w is angular velocity)

The Attempt at a Solution



Attempt 1

(a) F=umg, so u=F/Mag
I'm already skeptical about this, since I have the impression that F is exerted on the cart, not on the disk. So I should not be applying it to the disk.

(b) F=(Ma+Mb)a, find a
This in itself make no sense. If I draw a free body of the cart, the cart experiences zero resultant force. Moreover, N2L should not be applied to a non-rigid body and I should not add in 2 masses.

(c) Fr=I*alpha. Find alpha.

(d) a=r*alpha. Find a.

Something is really wrong with my understanding. Any instructions that can guide me to the correct direction will be appreciated. I will like to understand it in terms of classical Newton laws (2nd and 3rd law) and also freebody diagrams. More fundamentally, I will like to question if the only contact between a body and another is by friction (be it kinetic or static), when a force is applied to the formal body, isn't the force that is felt by the latter limited by friction? I also have an inkling this is about using of N2L in non-inertia frame.

Attempt 2
This attempt is based on 'friction force = 0 under rolling'.

(a) Same as attempt 1.
(b) F=mbab, find ab
(c) rF=I*alpha, find alpha
(d)
Due to rolling, arolling=r*alpha
Due to pseudo force in non-inertia frame, apseudo=F/ma
Acceleration in inertia frame = apseudo-arolling

In numerical answers, (b) 2ms-2 (c) 10rad/s2 (d) -3ms-2

I find this attempt slightly better, but it does not answer my previous doubt about whether the force on the disk is limited by the friction.

Attempt 3
I modified how I apply the psuedo force concept in part (c).

(c) Fpsuedo=maab
(since point of contact moves with cart)
rFpsuedo=I*alpha, find alpha

(d) Linear acceleration = 0
 
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  • #2
.In numerical answer, (b) 2ms-2 (c) 4rad/s2 (d) 0This attempt make more sense to me, and it also answers my doubt about whether the force on the disk is limited by the friction.
 

1. What is the difference between static and kinetic friction?

Static friction is the force that prevents an object from moving when a force is applied to it. It occurs when two surfaces are in contact but not moving relative to each other. Kinetic friction, on the other hand, is the force that opposes the motion of an object when it is already in motion. It occurs when two surfaces are in contact and moving relative to each other.

2. How does the mass of the ball affect its rolling motion on the cart?

The mass of the ball affects its rolling motion on the cart because it determines the amount of inertia the ball has. Inertia is the resistance of an object to change its state of motion, so a heavier ball will have more inertia and require more force to start and maintain its rolling motion on the cart.

3. What is the role of friction in the rolling motion of a ball on a cart?

Friction plays a crucial role in the rolling motion of a ball on a cart. It provides the necessary force to initiate and maintain the rolling motion. Without friction, the ball would simply slide on the surface of the cart rather than roll. Friction also helps to slow down the rolling motion of the ball, eventually bringing it to a stop.

4. How does the shape of the ball affect its rolling motion on the cart?

The shape of the ball can have a significant impact on its rolling motion on the cart. A perfectly round ball will have a more uniform distribution of mass, resulting in smoother rolling motion. On the other hand, a ball with an irregular shape may experience more resistance and have a more unpredictable rolling motion.

5. Can the rolling motion of a ball on a cart be affected by external factors?

Yes, the rolling motion of a ball on a cart can be affected by external factors such as the surface of the cart, the roughness of the ball, and the presence of any other forces acting on the system. For example, a bumpy or uneven surface can cause the ball to deviate from its intended path, while a strong wind may provide an additional force that affects the rolling motion.

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