Pulley w/ attached weights/Linear+Angular acceleration.

In summary, the linear acceleration of the 12 kg mass is -1.225 m/s^2 and the angular acceleration of the disks is -2.45 rad/s^2. The objects are moving in opposite directions, with the 12 kg mass accelerating downward and the 4 kg mass accelerating upward. The moment of inertia for the 4 kg disk is 4 kg m^2.
  • #1
Sulla
8
0
Determine the linear and angular acceleration of the disks.

A diagram is illustrated as having a small disk (radius 1m) with a rope hanging on its edge with a mass of 12 kg attached to it. Another pulley disk (radius 2m) surrounds the smaller and first one; it has a rope attached to a 4 kg mass.


This is what I have so far:

T1 - m1g = m1a1
= t1-117.6 = 12a1

m2g - T2 = m2a2
= 2 [39.2 - T2 = 4a2]
= 78.4 - 2T2=8a2

---
In order to find the accel., I calculated the velocity from the radii and distance using arc length.

d2= 2(theta)
d1= 1(theta)

d2/d1= 2(theta)
d2=2a1
v2=2v1
a2=2a1

a(tangential)= (alpha)r (alpha= angular accel.)
a2=2(alpha)
2a1=2(alpha)
(alpha) = a1

T1 - 177.6 = 12a1
78.4 - 2T2 = 8a2=8*2(alpha)=16(alpha)
+ 2T2 - T1 = 4a1
___________________________________

-39.2 = 32a1
a1= -1.225

a2= 2(-1.225)



Having done that...are the objects decelerating not accelerating? And have I completed all the steps correctly?
 
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  • #2
can you show the information more clearly? i can't think of the picture
 
  • #3
I hope this helps!
 

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  • #4
Sulla said:
Determine the linear and angular acceleration of the disks.

A diagram is illustrated as having a small disk (radius 1m) with a rope hanging on its edge with a mass of 12 kg attached to it. Another pulley disk (radius 2m) surrounds the smaller and first one; it has a rope attached to a 4 kg mass.


This is what I have so far:

T1 - m1g = m1a1
= t1-117.6 = 12a1

m2g - T2 = m2a2
= 2 [39.2 - T2 = 4a2]
= 78.4 - 2T2=8a2

---
In order to find the accel., I calculated the velocity from the radii and distance using arc length.

d2= 2(theta)
d1= 1(theta)

d2/d1= 2(theta) <<== theta?
d2=2a1 <<== d2 and a1 have different dimensions. This is impossible.
v2=2v1
a2=2a1

a(tangential)= (alpha)r (alpha= angular accel.)
a2=2(alpha) <<== a2 and alpha have different dimensions. This is impossible.

2a1=2(alpha)
(alpha) = a1 <<== a1 and alpha have different dimensions. This is impossible.
T1 - 177.6 = 12a1 <<== dimensions?

78.4 - 2T2 = 8a2=8*2(alpha)=16(alpha)
+ 2T2 - T1 = 4a1
___________________________________

-39.2 = 32a1
a1= -1.225

a2= 2(-1.225)



Having done that...are the objects decelerating not accelerating? And have I completed all the steps correctly?
You may have gotten the right answer, but you certainly did not do things "correctly". When you do not use units it makes it very difficult to follow your work.

If the system starts from rest it is speeding up. Whether the accelerations are positive or negative depends on the directions you chose to be positive when you set up your equations.
 
  • #5
Assuming that up and right (y) are postive and down and left are negative (x), so the answer would still remain negative. Would that indicate that the object on the left (12 kg) is moving upwards while the object on the right (4 kg) is moving downwards? If the acceleration of both are negative wouldn't that indicated that both are moving in the same direction?
 
  • #6
Sulla said:
Assuming that up and right (y) are postive and down and left are negative (x), so the answer would still remain negative. Would that indicate that the object on the left (12 kg) is moving upwards while the object on the right (4 kg) is moving downwards? If the acceleration of both are negative wouldn't that indicated that both are moving in the same direction?

In your original equations

T1 - m1g = m1a1
= t1-117.6 = 12a1

m2g - T2 = m2a2

you identified the 12kg mass as m1 and the 4kg mass as m2. Your first equation is written as positve upward. Your second equation is written as positive downward. This is a reasonable choice, since one mass will move upward and the other will move downward. Your negative answers for both accelerations just means that in fact the 12kg mass (m1) will accelerate downward while the 4kg mass (m2) will accelerate upward. The magnitude of the static (i.e., pulleys held in place) CCW torque, 12kg*1m*g, is greater than the magnitude of the static CW torque, 4kg*1m*g, so these directions are to be expected.

I don't see how you got your answer. Do the disks have mass? Is there a moment of inertia given?
 
  • #7
I apologise, I left out the fact that I = 4 kg m^2 .
 

1. How does a pulley with attached weights affect linear acceleration?

When a pulley with attached weights is used, the weights exert a downward force on the pulley. This downward force translates into a tension force on the rope or cable attached to the pulley. This tension force can then be used to accelerate an object in a linear direction, either by pulling the object directly or by using the tension force to counteract another force acting on the object.

2. Can a pulley with attached weights also affect angular acceleration?

Yes, a pulley with attached weights can also affect angular acceleration. When the weights on the pulley are not evenly distributed, the pulley will experience a torque force which can cause it to rotate. This rotation can then be used to accelerate an object in an angular direction, such as in the case of a wheel or axle.

3. How does the number of weights attached to a pulley affect acceleration?

The number of weights attached to a pulley can affect acceleration in a few ways. First, the total weight of the weights will determine the amount of force that can be exerted on the pulley, which in turn affects the amount of tension force that can be applied to an object. Additionally, the distribution of the weights can affect the direction and magnitude of the resulting torque force, which can impact angular acceleration.

4. What role does the mass of the weights play in acceleration?

The mass of the weights attached to a pulley can determine the amount of force that can be exerted on the pulley, as well as the amount of inertia the pulley will have when rotating. This can impact both linear and angular acceleration, as the greater the mass of the weights, the greater the force and inertia.

5. Are there any other factors that can affect the acceleration in a pulley system?

Yes, there are other factors that can affect acceleration in a pulley system. These include the friction between the pulley and the rope or cable, the angle at which the rope or cable is pulled, and the efficiency of the pulley system. Additionally, external forces acting on the object being accelerated, such as air resistance, can also impact the overall acceleration.

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