Rotational Dynamics: Confused about the tension in this string pulling a mass up an incline

  • #1
Sep
2
2
TL;DR Summary
I am confused why the solutions say that Mgsin(theta) is greater than tension. Isn't tension greater because the block is going up the incline? Or is it that in cases where the block is decelerating but still going up, the force of gravity will be greater?
Question

A wheel of radius r and moment of inertia I about its axis is fixed at the top of an inclined plane of inclination θ as shown in the figure. A string is wrapped round the wheel and its free end supports a block of mass M which can slide on the plane. Initially, the wheel is rotating at a speed of ω in a direction such that the block slides up the plane. How far will the block move before stopping?​


1714598075216.png

Suppose the deceleration of the block is a. The linear deceleration​

of the rim of the wheel is also a. The angular deceleration of the​

wheel is α=ar. I the tension in the string is T, the equations of​

motion are as follows:​

Mgsinθ−T=Ma (This is the part that I am confused about)​


and Tr=Iα=Iar​


Eliminating T from these equations​


Mgsinθ−Iar2=Ma​


giving a=Mgr2sinθI+Mr2​


I am confused why the solutions say that Mgsin(theta) is greater than tension. Isn't tension greater because the block is going up the incline? Or is it that in cases where the block is decelerating but still going up, the force of gravity will be greater?
 
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  • #2
Sep said:
Isn't tension greater because the block is going up the incline?
The block has positive velocity upwards, but that has nothing to do with the tension. Consider that if we cut the string the block would still keep coasting up the ramp for a while, until gravity brought it to a stop and started it back down.
 
  • Informative
Likes Sep
  • #3
Nugatory said:
The block has positive velocity upwards, but that has nothing to do with the tension. Consider that if we cut the string the block would still keep coasting up the ramp for a while, until gravity brought it to a stop and started it back down.
Oh, I understand now. The initial velocity. Thank you so much!
 
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Likes Nugatory and berkeman

FAQ: Rotational Dynamics: Confused about the tension in this string pulling a mass up an incline

1. What is rotational dynamics?

Rotational dynamics is the branch of physics that deals with the motion of rotating bodies and the forces that cause these rotations. It involves concepts such as torque, angular momentum, and the moment of inertia, which are essential for understanding how objects rotate and how they respond to applied forces.

2. How does tension in a string affect the motion of a mass on an incline?

The tension in the string plays a crucial role in determining the net force acting on the mass on the incline. It counteracts the component of gravitational force acting parallel to the incline, thus affecting the acceleration of the mass. If the tension is greater than the gravitational component, the mass will accelerate upwards; if less, it will accelerate downwards.

3. What forces should I consider when analyzing a mass on an incline with a tensioned string?

When analyzing the system, you should consider the gravitational force acting on the mass, the normal force perpendicular to the incline, the tension in the string, and any frictional forces that may be present. The gravitational force can be resolved into two components: one parallel to the incline and one perpendicular to it.

4. How do I calculate the tension in the string?

To calculate the tension in the string, you can use Newton's second law. First, draw a free-body diagram of the mass on the incline. Then, sum the forces acting on the mass in the direction of motion (parallel to the incline). Set up the equation using the net force, which includes the tension and the gravitational component, and solve for the tension.

5. What is the role of friction in this scenario?

Friction opposes the motion of the mass along the incline. If friction is present, it must be included in the free-body diagram and the equations of motion. The frictional force will reduce the net force acting on the mass, thus affecting the tension in the string and the overall acceleration of the system. If the incline is frictionless, the tension will be solely determined by the gravitational force and the angle of the incline.

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