Find accelerations- Angular and linear

In summary, the tension of the string is different depending on the direction the masses are pointing. The angular acceleration of the pulley is also different depending on the direction the masses are pointing.
  • #1
cbarker1
Gold Member
MHB
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Dear Every Body,

I need some help. First question: Is the tension of the string is the same or different?
A pulley of moment of inertia 2.7 kg [FONT=&quot]·
m2 is mounted on a wall as shown in the following figure. Light strings are wrapped around two circumferences of the pulley and weights are attached. Assume the following data: r1 = 47 cm,
r2 = 20 cm,
m1 = 1.0 kg,
and m2 = 1.9 kg.
[/FONT]

10-7-p-092.png
(a)
What is the angular acceleration in rad/$s^2$ of the pulley?


(b)
What is the linear acceleration (in m/s2) of the weights?
a1

a2

 
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  • #2
Hi Cbarker1,

Which formulas do you have that:
- relate torque to angular acceleration?
- relate torque to force?

Can we apply them?
 
  • #3
I like Serena said:
Hi Cbarker1,

Which formulas do you have that:
- relate torque to angular acceleration?
- relate torque to force?

Can we apply them?

$\Sigma \tau=I\alpha$
$\tau=Fr$
$\Sigma F=ma$
For the first hanging mass:
$m_1g-T=m_1a_1$
$T=m_1g-m_1a_1$
For the second hanging mass:
$T=m_2a_2+m_2g$
 
  • #4
Cbarker1 said:
$\Sigma \tau=I\alpha$
$\tau=Fr$
$\Sigma F=ma$
For the first hanging mass:
$m_1g-T=m_1a_1$
$T=m_1g-m_1a_1$
For the second hanging mass:
$T=m_2a_2+m_2g$

Good!
We also need that the speed of both masses match the speed that the wheel turns (where the masses are attached).
Can we also catch those in formulas?
 
  • #5
Do the mass have same tension?
 
  • #6
Cbarker1 said:
Do the mass have same tension?

If the system were in equilibrium, the corresponding torques would have to cancel.
Since they are at different distances, that means that in equilibrium the tensions have to be different.
So we can't assume that the tensions are the same, and we have to keep them like $T_1$ and $T_2$ instead of just $T$.
 
  • #7
I like Serena said:
If the system were in equilibrium, the corresponding torques would have to cancel.
Since they are at different distances, that means that in equilibrium the tensions have to be different.
So we can't assume that the tensions are the same, and we have to keep them like $T_1$ and $T_2$ instead of just $T$.
$$T_2=1.9a_2+1.9g$$
$$T_1=g-a_1$$
$$T_1r_1+T_2r_2=I\alpha $$
 
  • #8
Cbarker1 said:
$$T_2=1.9a_2+1.9g$$
$$T_1=g-a_1$$
$$T_1r_1+T_2r_2=I\alpha $$

Let's be careful with our signs.
The tensions that act on the masses both point upward yes?
Ideally we pick a direction up or down that we call positive and then stick to it.
How about picking 'up' to be 'positive' (it seems you already did)?

Which directions did you pick for $a_1$ and $a_2$?
Since we don't know yet if they are up or down, can we pick them both to be in the 'positive' direction?
If we do, there is a sign that we need to correct.

As for the angular acceleration, the torques oppose each other do they not?
That means one of them should have a minus sign.

Can we also include 2 more equations for the speeds of the masses that each must equal the speed of the wheel where the strings attach?
 
  • #9
I think \(\displaystyle a_1\) should be positive and \(\displaystyle a_2\) should be negative.
 
Last edited:
  • #10
Cbarker1 said:
I think \(\displaystyle a_1\) should be positive and \(\displaystyle a_2\) should be negative.

That's also fine.
A drawing would be nice then, since otherwise I have no clue which vector points where.
 
  • #11
here is the picture
 

FAQ: Find accelerations- Angular and linear

1. What is the difference between angular and linear acceleration?

Angular acceleration is the rate of change of angular velocity, which is the speed at which an object rotates around an axis. Linear acceleration is the rate of change of linear velocity, which is the speed at which an object moves in a straight line. In other words, angular acceleration relates to rotational motion, while linear acceleration relates to straight-line motion.

2. How do you calculate angular acceleration?

Angular acceleration can be calculated by dividing the change in angular velocity by the change in time. The formula for angular acceleration is: α = (ω2 - ω1) / (t2 - t1)

3. What are some real-life examples of angular acceleration?

Some common examples of angular acceleration include a spinning top, the rotation of a car's wheels, and the movement of a pendulum. Other examples include the rotation of a ball thrown in the air and the spinning of a figure skater during a routine.

4. Can angular acceleration be negative?

Yes, angular acceleration can be negative. A negative angular acceleration indicates that the object is slowing down its rotation. This can occur when an external force, such as friction, acts against the object's rotation.

5. How does angular acceleration relate to centripetal acceleration?

Angular acceleration and centripetal acceleration are closely related. Centripetal acceleration is the acceleration towards the center of a circular path, while angular acceleration is the rate of change of angular velocity. The formula for centripetal acceleration is: ac = (v2) / r, where v is the object's linear velocity and r is the radius of the circular path. This formula can be rewritten as ac = (ω2)r, which shows the relationship between centripetal acceleration and angular acceleration.

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