Block and pulley on movable incline

In summary, the conversation discusses the equations needed to solve for the acceleration of a string on an inclined plane. The speaker mentions that they were able to derive the force equations, but were unable to solve them to find the answer. They also mention the effect of tension on the pulley and the acceleration of the string in both the ground frame and the inclined frame. The speaker also questions if their force equations are correct and discusses the need for careful force balance in the problem.
  • #1
Aurelius120
251
24
Homework Statement
In the given figure if acceleration of Mass M with respect to ground is a then, find the acceleration of mass, m with respect to incline and with respect to the ground in terms of 'a'
Relevant Equations
T - Tcosα + Nsinα = Ma
mgsinα - T = ma
N = mgcosα
This was the question:
20230511_033251.jpg


I derived the equations as mentioned in the relevant equations.

But I could not solve the equations to find the answer. I realise with respect to inclined plane the acceleration must be a since string cannot slack. With respect to ground, the acceleration of incline is added.

However, I could solve the force equations to get required acceleration. Other answers seem to ignore the effect of tension on the pulley due to pulley and use the former method.

What is the acceleration of the string in the ground frame? In the frame of the incline the acceleration on either side of pulley is equal and string does not slack.

In the ground frame, the string on the side of the block will get an horizontal component of acceleration but the fixed end of the string will have same acceleration. This implies the string should slack which it does not. So where is my mistake?How do I solve the problems using the force equations I derived?
Are my force equations correct.
 
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  • #2
##N## doesn't equal ##mg\cos \alpha##, the wedge has a component of its acceleration opposite the direction of the normal force acting on the hanging mass. Imagine the wedge to be moving away (effectively falling out from underneath it) from the hanging mass. Likewise, I suspect you must also be more careful in your force balance on the hanging mass parallel to the slope. If you were on the ground you would see the hanging mass accelerating down the slope, and with the wedge.

A couple FBD's a preferable here to just writing down some equations
 
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  • #3
Aurelius120 said:
mgsinα - T = ma
The acceleration of the wedge has a component parallel to the slope. The sum of the real forces on the block parallel to the slope gives the acceleration of the block parallel to the slope in the ground frame, not in the wedge frame.
If you want to use the wedge frame you must add the inertial "virtual" force.
 

FAQ: Block and pulley on movable incline

What is a block and pulley on a movable incline system?

A block and pulley on a movable incline system is a classical mechanics setup where a block is placed on an inclined plane that can move, and the block is connected to a pulley system. This setup is used to study the dynamics of motion, forces, and friction, and it often involves analyzing the relative motion of the block and the incline, as well as the tension in the pulley system.

How do you analyze the forces acting on the block and the incline?

To analyze the forces, you need to consider both the gravitational forces acting on the block and the incline, the normal force, the frictional force (if any), and the tension in the string of the pulley. You typically start by drawing free-body diagrams for both the block and the incline, then apply Newton's second law to each body separately to set up equations of motion. These equations can then be solved simultaneously to find the unknowns such as acceleration, tension, and normal force.

What role does friction play in the block and pulley on a movable incline system?

Friction can significantly affect the motion of the block and the incline. If friction is present between the block and the incline, it will oppose the motion of the block relative to the incline. This frictional force must be included in the free-body diagram and the equations of motion. The coefficient of friction determines the magnitude of this force, and the friction can be either static or kinetic, depending on whether the block is stationary or moving relative to the incline.

How do you account for the motion of the incline in the analysis?

The motion of the incline must be considered because it affects the relative motion of the block. If the incline can move, you need to set up a coordinate system that takes into account the movement of both the block and the incline. The acceleration of the incline will influence the net forces acting on the block. By applying Newton's second law to both the block and the incline, you can derive coupled equations that describe the system's dynamics and solve for the accelerations of both the block and the incline.

What are some common assumptions made in solving problems involving a block and pulley on a movable incline?

Common assumptions include neglecting air resistance, assuming the pulley is frictionless and massless, and sometimes assuming the string is inextensible and massless. Additionally, if friction is considered, it is often assumed to be constant and characterized by a known coefficient of friction. These simplifications make the mathematical analysis more tractable and allow for a clearer understanding of the fundamental principles at play.

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