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Consider an in-extensible string passing over an ideal pulley connected to a block of mass $m$ as shown:

The ends are pulled with a force $F$

Am I correct in saying that the block will experience a reaction force $\sqrt{2}F$? The forces acting are perpendicular to each other and the support of the pulley is inclined at an angle of $45^\circ$ with horizontal.

In other words can the forces acting on the string be vectorially added to find the resultant reaction force exerted by pulley?

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    $\begingroup$ I encountered this specific scenario while solving an entirely different question. That is the reason I havent given any context. Im not sure about why this is tagged as a homework question. This is a purely conceptual doubt $\endgroup$
    – newbie105
    Commented Jan 2, 2021 at 16:27
  • $\begingroup$ If you replace the block and pulley with a bigger block, with the same center of mass, with two strings attached near the corner (at the same places that the rope comes into contact with the pulley) how would it be different? $\endgroup$
    – Alex K
    Commented Jan 9, 2023 at 1:35

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Yes, this is correct. However, the pulley will also experience a reaction force from the connection to the block.

Two things to note:

  1. Since this is a pulley, there is also torque, since the forces are not acting on the center of mass (in this case the torques cancel).
  2. As soon as the pulley starts moving, your analysis will change, because the force change direction.
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  • $\begingroup$ Can you explain why the directions of the applied forces change? Wouldn't they be acting horizontally and vertically just like before? $\endgroup$
    – newbie105
    Commented Jan 2, 2021 at 16:49
  • $\begingroup$ @newbie105 It depends on how the ropes are attached. If the ropes are fixed at the point, the rope will have a different angle, and therefore also the force. $\endgroup$
    – Bernhard
    Commented Jan 2, 2021 at 18:18
  • $\begingroup$ Can the downvoter explain their vote? $\endgroup$
    – Bernhard
    Commented Jan 3, 2021 at 8:47
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    $\begingroup$ @Buraian I would consider it a normal force, not a friction force. $\endgroup$
    – Bernhard
    Commented Jan 3, 2021 at 8:48
  • $\begingroup$ Your answer would be complete if you also address the more general question of the possibility of vector addition for any angle. I would also like to have a proof for the same. $\endgroup$
    – newbie105
    Commented Jan 3, 2021 at 12:40
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It depends if you want to consider the pulley massless or not. With mass (and rotational inertia) the two forces you drew aren't equal, and this will modify the diagram. The significance of this is that the support arm of the pulley would experience not only an inward force in the direction you drew, but also a lateral one that'll produce a torque that in the real world could snap off the support arm, for instance... if we assume the arm is oriented like your pink arrow.

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Yes, you are right. The tension in the thread is equal to the force applied by the external agent.

In this case, $ T = F $

and $ \because \theta = 90^{\circ} $ between both the force vectors, $ T_{net}= \sqrt{2}\,T = \sqrt2F $.

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