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All attachments go through a moderation queue and must be approved by a mentor beofre they are accessible. This one hasn't been approved just yet.Physics Monkey said:Hi don,
I can't see the attachment for some reason,
...or write the cosine in terms of the sine, and then do the above....but if I understand the problem correctly, it looks like you did everything fine. You now simply need to solve for [tex] \theta [/tex] using the equation
[tex] 2 \sin{\theta} - 2 \mu_k \cos{\theta} - 1 = 0 [/tex].
Unfortunately, the differential equation you have written is not equivalent to the equation you derived from Newton's Laws. This is because the differential has a solution that will be valid for all [tex] \theta [/tex] (and will be exponential) while you are looking for a particular value of [tex] \theta [/tex]. How do you solve for [tex] \theta [/tex] then? Here is hint: try isolating [tex] \sin{\theta} [/tex] and squaring both sides.
I think it's one of those problems that turns out to reveal more neat stuff than the teacher expected. At angles between the two limiting cases, the blocks would, as you said above, decelerate from the initial velocity... and come to rest (rather than change direction, because at the instant that v=0 they'd experience static friction which will then take over). So, in the steady state, they'd have a constant velocity.Physics Monkey said:Quite true, Gokul. Hopefully the down angle is sufficient for his problem.
However, I do think there is a slight mistake with your second point. The force of kinetic friction is independent of speed, it doesn't just match the forces the way static friction does. So if one were to replace [tex] \mu_k [/tex] with [tex] \mu_s [/tex] and then solve the "about to move down" and the "about to move up" cases, the block could sit at rest at any angle in between these two extreme cases owing to the fact that static friction will be whatever it takes to just keep the blocks at rest. However, if the block is initially moving, then there are only two angles, the up angle and the down angle, where the force of kinetic friction balances the other forces. What do you think?
You bet !BTW: Thanks for Homework Helper nod.
The Atwood's machine problem is a physics problem that involves a system of two masses connected by a string or pulley. It is named after George Atwood, an English mathematician who first described it in the late 18th century. The problem is usually set on an inclined plane and involves finding the acceleration of the masses and the tension in the string.
The Atwood's machine works by utilizing the principles of Newton's laws of motion. The two masses are connected by a string or pulley, and as one mass moves down the inclined plane, the other mass moves up. This creates a tension in the string, which causes the masses to accelerate. The direction and magnitude of the acceleration can be determined by using equations of motion and considering the forces acting on the system.
The key components of the Atwood's machine problem are the two masses, the string or pulley connecting them, and the inclined plane. Other factors that can affect the problem include the angle of the incline, the mass of the string, and any external forces acting on the system.
The Atwood's machine problem is solved by using equations of motion and applying the principles of Newton's laws of motion. The first step is to draw a free-body diagram of the system and identify all the forces acting on the masses. Then, using the equations of motion, the acceleration and tension in the string can be calculated. Finally, the solution can be verified by checking if the forces and accelerations are consistent with the given conditions and constraints.
The Atwood's machine problem has various real-world applications, including elevator systems, weightlifting machines, and cranes. In elevators, the motor pulls on a cable that is connected to a counterweight, similar to the Atwood's machine. Weightlifting machines also use a similar mechanism to provide resistance for muscle training. Cranes use a combination of pulleys and weights to lift heavy objects, which is another example of the Atwood's machine problem in action.