How Does a Lift's Acceleration Affect the Pulley System Dynamics?

In summary: So, we agree that the magnitude of the effective gravity measured in the lift is ##g+a_0##.In summary, the problem involves two blocks attached to a pulley with different radii and a moment of inertia, and supported by a bearing with negligible friction. The first part of the problem asks for the magnitude of the angular acceleration of the system with given masses. The second part involves the system being inside a lift with an acceleration upwards. In this case, the equations for dynamic motion involve adding the acceleration of the lift to the gravitational acceleration. The equations for the rotation motion of the pulley remain the same. The effective gravity measured in the lift would be the
  • #1
Victoria_235
13
0

Homework Statement


Two blocks are attached by strings of negligible mass to a pulley with two radii R1=0.06 m and R2=0.08 m. The strings are wrapped around their respective radios so that the masses can move either up or down. The pulley has a moment of inertia I=0.0023 kg
char01.png
m2 , and is supported by a bearing with negligible friction.
1)If block 1 has a mass of 0.6 kg and block 2 has a mass of 0.5 kg, what is the magnitude of the angular acceleration of the system?
2) Idem, but if the system is moving up with an acceleration a_0.
(My problem is part two.)

Homework Equations


I have solved the part 1) without problems.

The equations would be:

\begin{equation}
T_1 - m_1g = m_1a_1 \\
m_1g - T_2 = m_2a_2\\

Where:\\
a_1 = \alpha R_1 \\
a_2 = \alpha R_2\\
\end{equation}
For the rotation motion of the pulley:
\begin{equation}
\sum M = I \alpha\\
\sum M = F \times r\\

T_2 R_2 -T_1R_1 = I \alpha\\
\end{equation}
Solving the system equation, I have obtained: \\
\begin{equation}
T_1 = 6.06 N\\
T_2 =4.69N\\
\alpha = 5.117 rad/s²\\
\end{equation}
Where $\alpha$ is the angular acceleration.

My problem comes in the part 2. Ths system inside a lift with an acceleration up.
My attempt solution...

The Attempt at a Solution



The only thing I think I can do is add to $\vec{g}$ the acceleration $\vec{a_0}$.\\
The equations in the lift would be:

\begin{equation}
T_1 - m_1(\vec{g}+\vec{a_0})= m_1(\vec{a_1}+\vec{a_0}) \\

m_1(\vec{g}+\vec{a_0}) - T_2 = m_2(\vec{a_2}-\vec{a_0})\\
\end{equation}

(In here I am not sure if I have to add a_0 and subtract it, depending if the mass is going down or up. Also if I have to add a_0 to g in the term:$$m_1(\vec{g}+\vec{a_0})$$ and $$m_2(\vec{g}+\vec{a_0})$$.For the rotation motion of the pulley:\\
\begin{equation}

T_2 R_2 -T_1R_1 = I \alpha\\

\end{equation}

Should I add the $\vec{a_0}$ in the rotation equations?
PS: I am sorry for my poor LaTeX, I will try to improve it.
 

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  • #2
You can always go to the accelerated frame. In this frame the effective gravitational field will indeed be ##\vec g -\vec a_0## (note that ##\vec g## and ##\vec a_0## are directed in opposite directions, the first down and the second up) and the problem will be exactly equivalent to the first, but with a different gravitational field.
 
  • #3
So, I understand that the equations (4), about dinamic motion, would be:

\begin{equation}
T_1 -m_1 (\vec{g}- \vec{a_0}) =m_1a_1\\
m_2 (\vec{g}- \vec{a_0}) -T_2 =m_2a_2\\
\end{equation}
When the lift is going up. Because someone inside the lift would fell the gravity harder if the lift is going up, for this reason I don't understand why should be $$(\vec{g}+\vec{a_0})$$, as you say...
 
  • #4
Victoria_235 said:
So, I understand that the equations (4), about dinamic motion, would be:

\begin{equation}
T_1 -m_1 (\vec{g}- \vec{a_0}) =m_1a_1\\
m_2 (\vec{g}- \vec{a_0}) -T_2 =m_2a_2\\
\end{equation}
When the lift is going up. Because someone inside the lift would fell the gravity harder if the lift is going up, for this reason I don't understand why should be $$(\vec{g}+\vec{a_0})$$, as you say...

##\vec{g}## and ##\vec{a_0}## are vectors. In general, the effective gravity will be ##\vec{g_{eff}} = \vec{g} - \vec{a_0}##.

If we put in some numbers. Let ##\vec{g} = -10 m/s^2## and ##\vec{a_0} = + 2m/s^2##, then ##\vec{g_{eff}} = \vec{g} - \vec{a_0} = -12m/s^2##.

But, in this case, in terms of magnitudes: ##g_{eff} = g+a_0 = 12m/s^2##
 

Related to How Does a Lift's Acceleration Affect the Pulley System Dynamics?

1. How does a pulley system work in a lift?

A pulley system in a lift uses a combination of fixed and movable pulleys to lift and lower the lift car. The fixed pulleys are attached to the lift car and the lift shaft, while the movable pulleys are attached to the lift car and the counterweight. As the lift car moves up, the counterweight moves down, and vice versa. The pulleys work together to reduce the amount of force needed to lift the car, making it easier for the lift motor to operate.

2. What are the advantages of using a pulley system in a lift?

The main advantage of using a pulley system in a lift is that it reduces the amount of force needed to lift the car, making it more efficient and less taxing on the lift motor. Additionally, the use of a counterweight helps to balance the weight of the lift car, minimizing the risk of the lift becoming stuck or imbalanced. Pulley systems are also relatively simple and cost-effective compared to other lifting mechanisms.

3. How does the number of pulleys affect the lift's performance?

The number of pulleys used in a lift affects its performance by changing the mechanical advantage of the system. Generally, the more pulleys used, the easier it is to lift the car, as each additional pulley reduces the required force by half. However, using too many pulleys can also increase the amount of friction and decrease the efficiency of the system.

4. What are some common maintenance issues with pulley systems in lifts?

Some common maintenance issues with pulley systems in lifts include wear and tear on the pulleys and cables, misalignment of pulleys, and insufficient lubrication. These issues can cause the lift to operate less efficiently or even become stuck. Regular maintenance and inspection of the pulley system can help prevent these issues and ensure the safe and smooth operation of the lift.

5. Are there different types of pulley systems used in lifts?

Yes, there are different types of pulley systems used in lifts, including single rope, double rope, and compound pulley systems. Single rope systems use one continuous cable, while double rope systems use two separate cables. Compound pulley systems combine multiple fixed and movable pulleys to further increase the mechanical advantage. The type of pulley system used in a lift will depend on factors such as the weight capacity and desired speed of the lift.

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