Q62P - University Physics with Modern Physics

Question

a worker lifts a weight w  by pulling down on a rope with a force $\overrightarrow{\mathbf{F}}$ . The upper pulley is attached to the ceiling by a chain, and the lower pulley is attached to the weight by another chain. Draw one or more free-body diagrams to find the tension in each chain and the magnitude of $\overrightarrow{\mathbf{F}}$ , in terms of w , if the weight is lifted at constant speed. Assume that the rope, pulleys, and chains have negligible weights.

                                                   

Step-by-Step Solution

Answer

The tension in both the chains is  w and the magnitude of $\overrightarrow{F}$  is $\frac{w}{2}$ .

Step 1: Identification of the given data

The given data can be listed below as:

• The weight lifted by the worker is w .
• The rope has been pulled with a force of  $\overrightarrow{F}$.

Step 2: Significance of the weight

The weight is described as the force that acts on an object because of gravity. It also helps the earth to attract masses into its center.

Step 3: Determination of the tension and the force

The free-body diagram of the system has been drawn below:

 

In the above diagram, the weight w is acting in the downwards direction along with the force $\overrightarrow{F}$. The velocity v  is acting in the upwards direction.

 

The diagram representing the forces acting on the weight-

 

 

In the above diagram,  W is the weight of the body and the tension $T_1$ is acting on the chain in upwards direction.

From the diagram of the weight, the equation of the tension on the chain is expressed as:

 

$$\begin{gathered} T_1-w=0 \\ T_1=0 \end{gathered}$$ 

 

Here, $T_1$ is the tension in the chain and it is equal and opposite to that of the weight (W) hanging from the chain.

The diagram of the movable pulley has been drawn below:

 

 

 

The weight  w is acting in the downwards direction and the tension T is acting in the string in upwards direction.

 

From the diagram of the movable pulley, the equation of the tension in the left pulley is expressed as:

 

$$\begin{gathered} T+T=T_1 \\ 2T=W \end{gathered}$$  

 

Here,  T is described as the tension acting on the string.

Above equation can also be written as:

 $T=\frac{W}{2}\hspace{0.33em}\hspace{0.33em}\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \left(1\right)$ 
The diagram of the fixed pulley has been drawn below:

Here, T  is the tension in the string and $T_2$  is the tension in the chain through which pulley is fixed from the roof.

From the above diagram, the equation of force is given as-                   

$$\begin{gathered} T_2=T+T \\ {T}_2=2T\hspace{0.33em}\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \left(2\right) \end{gathered}$$  

Using equation (1) , 

$$\begin{gathered} T_2=2\times \frac{W}{2} \\ =W \end{gathered}$$  

Here, $T_2$ is the tension in the chain through which pulley is fixed from the roof.

 

The diagram of the point of application of force F,

Here, T is the tension in the string and F is the magnitude of force applied.

 

From the above diagram of point of application of force and using equation (1), the equation of the Force  (F)  is given as:

 

 $$\begin{gathered} F=T \\ F=\frac{w}{2} \end{gathered}$$ 

 

Here, $T_2$ is the tension in the right pulley.

 

Thus, the tension in both the right and the left chain is  w and the magnitude of  $\overrightarrow{F}$ is $\frac{w}{2}$.