Two questions concerning mechanics

[Pseudo Statistic]

Hey,
I have a question about certain conditions in Newtonian mechanics...
Let's say you had an incline, with a pulley at the edge of the top, frictionless, let's keep the string & pulley masses negligible, and there's a light string connecting an object hanging, and an object on the incline...
I know that all of our assumptions that tension is equal is true, but how is the tension equal throughought? Isn't it accelerating a different value on both sides? I know I'm probably asking a stupid question, but if the two masses are different (one is 3kg, one is 9kg, for instance) the acceleration on one due to the tension will be more than the other, wouldn't it? Or would it not?
Can someone explain this to me with proof behind the explanation? (Again, apologize if this is a stupid question)

Also, let's say a projectile was thrown with some initial velocity. At its highest point, when we consider its potential energy, (Let's say it has a constant horizontal velocity, i.e. it was given that initial velocity at an angle in the first place) do we consider it by considering both the horizontal potential and kinetic energy as well as the vertical potential and kinetic energy, (by using vector addition and the like) or do we consider them to be two different things and ignore the horizontal direction?
Can someone explain this with proof and perhaps an example?

Thanks a lot for any responses.

 

[Doc Al]

Hey,
I have a question about certain conditions in Newtonian mechanics...
Let's say you had an incline, with a pulley at the edge of the top, frictionless, let's keep the string & pulley masses negligible, and there's a light string connecting an object hanging, and an object on the incline...
I know that all of our assumptions that tension is equal is true, but how is the tension equal throughought? Isn't it accelerating a different value on both sides? I know I'm probably asking a stupid question, but if the two masses are different (one is 3kg, one is 9kg, for instance) the acceleration on one due to the tension will be more than the other, wouldn't it? Or would it not? Can someone explain this to me with proof behind the explanation? (Again, apologize if this is a stupid question)

Realize that as long as the string does not stretch, both masses are constrained to move together with the same speed and acceleration. What determines the acceleration of each mass is not merely the tension (which is the same throughout since the string is massless) but the net force on each mass, which includes gravity.

Also, let's say a projectile was thrown with some initial velocity. At its highest point, when we consider its potential energy, (Let's say it has a constant horizontal velocity, i.e. it was given that initial velocity at an angle in the first place) do we consider it by considering both the horizontal potential and kinetic energy as well as the vertical potential and kinetic energy, (by using vector addition and the like) or do we consider them to be two different things and ignore the horizontal direction?
Can someone explain this with proof and perhaps an example?

Potential energy depends only on the vertical position (height), so a "horizontal" PE makes no sense. Near the Earth the formula for gravitational PE is mgy, where y is measured from some arbitrary reference point.

KE energy depends on speed, not direction. But since $v^2 = v_x^2 + v_y^2$, it's OK to think of the KE as associated with the vertical and horizontal components of velocity separately, if it helps you. At the top of the projectile's motion, the only speed is horizontal so ${KE} = 1/2 m v^2 = 1/2 m v_x^2$.

Both PE and KE are scalars, not vectors, so you just add them like ordinary numbers. Using vector addition makes no sense, since energy does not have a direction.

 

[charng]

Hello,

Thank you for your questions regarding Newtonian mechanics. I will do my best to answer them with explanations and proof.

Firstly, in regards to the tension in the string connecting the hanging object and the object on the incline, it is true that the two masses will experience different accelerations due to the tension. This is because the acceleration of an object is directly proportional to the net force acting on it, and the net force is determined by the mass of the object. In this case, the hanging object with a mass of 3kg will experience a greater acceleration than the object on the incline with a mass of 9kg, assuming the same tension is applied to both objects. However, the tension in the string will still be equal throughout because of Newton's Third Law of Motion, which states that for every action, there is an equal and opposite reaction. The tension in the string is the action force on each object, and according to this law, it will be equal for both objects. This can be proven by using Newton's Second Law of Motion, which states that the net force on an object is equal to its mass multiplied by its acceleration. By setting the net force equations for both objects equal to each other and solving for the acceleration, you will see that the tension is equal on both sides of the string.

As for your second question about a projectile at its highest point, the potential energy of the projectile will only be affected by its vertical position. This is because potential energy is defined as the energy an object has due to its position or state, and in this case, the horizontal position does not change at the highest point. Therefore, only the vertical potential energy will be considered. However, the kinetic energy of the projectile will still have both horizontal and vertical components, and they can be added using vector addition to determine the total kinetic energy of the projectile.

I hope this helps to answer your questions. If you need further clarification or have additional questions, please don't hesitate to ask. Science is all about asking questions and seeking answers, so there is no such thing as a stupid question. Keep exploring and learning!