1. Two blocks are connected by a massless rope as shown below. The mass of the block on the table is m, and the hanging mass is m₂. The table and the pulley are frictionless. a₁ m₁ y L X m₂ Subpart 1: Draw FBDs In your notebook, draw free body diagrams for m, and m₂ using the template as shown below. The forces acting on the system are weights of the blocks, m₁g, and m₂g, the tension in the string T and the normal reaction N₁ of the table on m₁. 111 a₂ m2

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Chapter1: Units, Trigonometry. And Vectors
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**Problem Statement**

1. Two blocks are connected by a massless rope as shown below. The mass of the block on the table is \( m_1 \) and the hanging mass is \( m_2 \). The table and the pulley are frictionless.

*Diagram Explanation:*
- The diagram shows two blocks connected by a rope that runs over a pulley. 
- Block \( m_1 \) is on a horizontal surface, and block \( m_2 \) is hanging off the edge.
- The block \( m_1 \) experiences acceleration \( \vec{a_1} \) horizontally.
- The block \( m_2 \) experiences acceleration \( \vec{a_2} \) vertically downward.

**Subpart 1: Draw FBDs**

In your notebook, draw free body diagrams for \( m_1 \) and \( m_2 \) using the template as shown below. The forces acting on the system are weights of the blocks, \( m_1g \) and \( m_2g \), the tension in the string \( T \) and the normal reaction \( N_1 \) of the table on \( m_1 \).

*Free Body Diagram Template:*
- A coordinate system with axes labeled \( x \) (horizontal) and \( y \) (vertical).
- Represent \( m_1 \) and \( m_2 \) using circles and show vectors for forces such as weight, tension, and normal force.

**Subpart 2: Set up Newton's Second Law Equations**

*Instructions:*
- Apply Newton's Second Law to each block separately.
- For \( m_1 \), consider horizontal forces (tension and acceleration).
- For \( m_2 \), consider vertical forces (tension, weight, and acceleration).
Transcribed Image Text:**Problem Statement** 1. Two blocks are connected by a massless rope as shown below. The mass of the block on the table is \( m_1 \) and the hanging mass is \( m_2 \). The table and the pulley are frictionless. *Diagram Explanation:* - The diagram shows two blocks connected by a rope that runs over a pulley. - Block \( m_1 \) is on a horizontal surface, and block \( m_2 \) is hanging off the edge. - The block \( m_1 \) experiences acceleration \( \vec{a_1} \) horizontally. - The block \( m_2 \) experiences acceleration \( \vec{a_2} \) vertically downward. **Subpart 1: Draw FBDs** In your notebook, draw free body diagrams for \( m_1 \) and \( m_2 \) using the template as shown below. The forces acting on the system are weights of the blocks, \( m_1g \) and \( m_2g \), the tension in the string \( T \) and the normal reaction \( N_1 \) of the table on \( m_1 \). *Free Body Diagram Template:* - A coordinate system with axes labeled \( x \) (horizontal) and \( y \) (vertical). - Represent \( m_1 \) and \( m_2 \) using circles and show vectors for forces such as weight, tension, and normal force. **Subpart 2: Set up Newton's Second Law Equations** *Instructions:* - Apply Newton's Second Law to each block separately. - For \( m_1 \), consider horizontal forces (tension and acceleration). - For \( m_2 \), consider vertical forces (tension, weight, and acceleration).
## Newton's Second Law Analysis

### (i) Newton’s Second Law for \( m_1 \)

In terms of weights of the blocks \( m_1g \) and \( m_2g \), the tension in the string \( T \) and the normal reaction \( N_1 \) of the table on \( m_1 \) can be written as (using the coordinate axes for signs of different forces):

\[
\sum F_x = \underline{\hspace{3cm}} = m_1 a_{1,x}
\]

\[
\sum F_y = \underline{\hspace{3cm}} = m_1 a_{1,y}
\]

### (ii) Newton’s Second Law for \( m_2 \)

In terms of weights of the blocks \( m_1g \) and \( m_2g \), the tension in the string \( T \) and the normal reaction \( N_1 \) of the table on \( m_1 \) can be written as:

*Note: The motion of \( m_2 \) is along the y-direction only.*

\[
\sum F_y = \underline{\hspace{3cm}} = m_2 a_{2,y}
\]

### (iii) Relation Between \( \vec{a_1} \) and \( \vec{a_2} \)

How are \( \vec{a_1} \) and \( \vec{a_2} \) related? What are the directions of \( \vec{a_1} \) and \( \vec{a_2} \)?

**Magnitudes of \( \vec{a_1} \) and \( \vec{a_2} \):**

A. \( a_1 \) is greater than \( a_2 \)  
B. \( a_1 \) is less than \( a_2 \)  
C. \( a_1 \) is equal to \( a_2 \)  

**Directions of \( \vec{a_1} \) and \( \vec{a_2} \):**

A. \( \vec{a_1} \) is to the right and \( \vec{a_2} \) is up  
B. \( \vec{a_1} \) is to the right and \( \vec{a_2} \) is
Transcribed Image Text:## Newton's Second Law Analysis ### (i) Newton’s Second Law for \( m_1 \) In terms of weights of the blocks \( m_1g \) and \( m_2g \), the tension in the string \( T \) and the normal reaction \( N_1 \) of the table on \( m_1 \) can be written as (using the coordinate axes for signs of different forces): \[ \sum F_x = \underline{\hspace{3cm}} = m_1 a_{1,x} \] \[ \sum F_y = \underline{\hspace{3cm}} = m_1 a_{1,y} \] ### (ii) Newton’s Second Law for \( m_2 \) In terms of weights of the blocks \( m_1g \) and \( m_2g \), the tension in the string \( T \) and the normal reaction \( N_1 \) of the table on \( m_1 \) can be written as: *Note: The motion of \( m_2 \) is along the y-direction only.* \[ \sum F_y = \underline{\hspace{3cm}} = m_2 a_{2,y} \] ### (iii) Relation Between \( \vec{a_1} \) and \( \vec{a_2} \) How are \( \vec{a_1} \) and \( \vec{a_2} \) related? What are the directions of \( \vec{a_1} \) and \( \vec{a_2} \)? **Magnitudes of \( \vec{a_1} \) and \( \vec{a_2} \):** A. \( a_1 \) is greater than \( a_2 \) B. \( a_1 \) is less than \( a_2 \) C. \( a_1 \) is equal to \( a_2 \) **Directions of \( \vec{a_1} \) and \( \vec{a_2} \):** A. \( \vec{a_1} \) is to the right and \( \vec{a_2} \) is up B. \( \vec{a_1} \) is to the right and \( \vec{a_2} \) is
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