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

AI Thread Summary
In the discussion about a block being pulled up an incline by a wheel, the key point revolves around the relationship between tension, gravitational force, and the block's motion. The equations of motion indicate that while the block is moving upwards, the gravitational force component acting down the incline (Mgsinθ) is greater than the tension in the string (T). This is because, even though the block has a positive velocity upwards, it is decelerating due to gravity. The confusion arises from the assumption that tension should be greater when the block is moving up, but the deceleration caused by gravity plays a crucial role. Understanding that the block can still coast upwards momentarily before gravity takes over clarifies the dynamics at play.
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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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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.
 
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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