Hanging mass + acceleration

In summary, the cart is accelerating to the right with an acceleration of 13 m/s/s and the tension force in the string is 8.3 N. This is determined by setting up equations for the net forces along the x and y axes, taking into account the fictive force needed in a non-inertial frame of reference. The acceleration of the cart is equal to the acceleration of the hanging mass.
  • #1
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A 50kg weight hangs from the ceiling of a cart that is accelerating horizontally. While the cart is accelerating, the weight swings to the left so that the string makes an angle of 37 degree with the horizontal ceiling. a) What is the size and direction of the acceleration of the cart? b) What is the size of the tension force in the string?

Fnet = ma
Fw = mg
Ftx = Ftcos37
Fty = Ftsin37

So first I tried solving for the Fw using Fw = mg. After finding that I assumed that Fw = Ft and used it to solve for Ftx, but then I'm stuck and have no idea what I'm doing.
 
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  • #2
Such problems are often being observed from the reference frame fixed to the cart. Since the cart is accelerating, such reference frame is non-inertial. 1. and 2. Newton's laws are not valid there, but we can still use them (just formally, in order to have a mathematical description of the problem).

From such frame you can see that the body you got is hanging from the ceiling forming a certain angle (it is at rest!). Now, only two real forces on the body are gravitational force and string tension. But, according to 1. Newton's net force on the body must be zero for body to be at rest. So we introduce one fictive force (inertial force). Its magnitude is [tex]F_{i}=ma[/tex], where [tex]a[/tex] is acceleration of the cart. This force is always pointing in opposite direction of the cart acceleration. (Remember: this force actually doesn't exist!)

Now, only thing you need to do is to represent each of the forces you have (gravitational, tension and fictive force) along the x and y axis. Net forces along those axes must be zero.

So, you have these equations ([tex]T[/tex] is the tension force and [tex]\alpha[/tex] is the angle):

[tex]T \sin \alpha - ma = 0[/tex] (x axis, right is taken as positive direction)

[tex]T \cos \alpha - mg = 0[/tex] (y axis, up taken as the positive direction)

From there, you can easily obtain magnitude of the acceleration and tension force.
 
  • #3
Ok, so I understand that when the cart is accelerating, the hanging mass is at rest and what you propose make sense. So for this case, is the Ft equals to the Fw since the object is at rest? And I don't get this fictive force thing =|.
Btw, the answers are:
a) 13 m/s/s to the right
b) 8.3n

EDIT: Sorry I read the question wrong, the weight is only .50 kg!
 
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  • #4
Fictive force is here just to eliminate the effect of other two forces so that the body is at rest. You can also observe the problem without fictive force. Just remember that you are dealing with vectors.

Then you look at the net forces along each of the axes:

[tex]ma_{x}=T \sin \alpha[/tex] ([tex]a_{x}[/tex] is acceleration along x axis)

[tex]ma_{y}=T \cos \alpha - mg[/tex] ([tex]a_{y}=0[/tex] is acceleration along y axis)

From these two equations you get

[tex]a_{x}=g \tan \alpha[/tex]

[tex]T=\frac{mg}{\cos \alpha}[/tex]

where [tex]\alpha = 90^{\circ}-37^{\circ}=53^{\circ}[/tex] (I messed up the angle a bit because I didn't read the text of the problem well, sorry.) [tex]a_{x}[/tex] is the acceleration you are looking for (the acceleration of the body looking from the inertial frame of the Earth, not from the cart). Acceleration of the body is equal to the acceleration of the cart (again, if you look from outside the cart).

Sorry if I confused you with too much theory, but I think it is important to know what's going on.
 
  • #5
Thank you! I understand now.
 

1. How does the mass of a hanging object affect its acceleration?

The mass of a hanging object has a direct impact on its acceleration. According to Newton's Second Law of Motion, the greater the mass of an object, the more force is required to accelerate it. Therefore, a heavier hanging mass will experience a lower acceleration compared to a lighter hanging mass.

2. What is the relationship between the acceleration of a hanging mass and the force applied?

The acceleration of a hanging mass is directly proportional to the force applied to it. This means that if the force applied to the hanging mass increases, its acceleration will also increase. This relationship is described by Newton's Second Law of Motion, which states that force equals mass times acceleration (F=ma).

3. How does the length of the string or rope affect the acceleration of a hanging mass?

The length of the string or rope has a minimal effect on the acceleration of a hanging mass. As long as the string or rope is taut and not stretched, the acceleration of the hanging mass will be the same regardless of its length. This is because the force of gravity acting on the mass remains constant, and the mass of the string or rope is negligible compared to the hanging mass.

4. Does the angle of the string or rope affect the acceleration of a hanging mass?

The angle of the string or rope does not affect the acceleration of a hanging mass. As long as the string or rope is taut and not stretched, the acceleration of the hanging mass will remain the same regardless of the angle. This is because the force of gravity acting on the mass remains constant and is always directed towards the center of the Earth.

5. How does air resistance impact the acceleration of a hanging mass?

Air resistance can have a significant impact on the acceleration of a hanging mass. Air resistance is a force that opposes the motion of an object through air. As the hanging mass falls, it will experience air resistance, which will decrease its acceleration. The amount of air resistance depends on the speed and surface area of the object. In a vacuum, where there is no air resistance, the hanging mass will experience a greater acceleration.

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