Solve for a, T1, T2 in terms of the other variables. M1a=m1g-T1, 1/2Ma=T1-T2, m2a=T2-m2g

To solve for \( a \), \( T_1 \), and \( T_2 \) in terms of the other variables, let's proceed step by step. The system of equations is:

\[
\text{(1) } M_1a = m_1g - T_1
\]
\[
\text{(2) } \frac{1}{2}Ma = T_1 - T_2
\]
\[
\text{(3) } m_2a = T_2 - m_2g
\]

### Step 1: Solve for \( T_1 \) from Equation (1)
From Equation (1):
\[
T_1 = m_1g - M_1a
\]

### Step 2: Solve for \( T_2 \) from Equation (3)
From Equation (3):
\[
T_2 = m_2a + m_2g
\]

### Step 3: Substitute \( T_1 \) and \( T_2 \) into Equation (2)
Substituting \( T_1 = m_1g - M_1a \) and \( T_2 = m_2a + m_2g \) into Equation (2):
\[
\frac{1}{2}Ma = (m_1g - M_1a) - (m_2a + m_2g)
\]
Simplify the equation:
\[
\frac{1}{2}Ma = m_1g - M_1a - m_2a - m_2g
\]
\[
\frac{1}{2}Ma + M_1a + m_2a = m_1g - m_2g
\]
Factor out \( a \) on the left side:
\[
\left(\frac{1}{2}M + M_1 + m_2\right)a = m_1g - m_2g
\]
Solve for \( a \):
\[
a = \frac{m_1g - m_2g}{\frac{1}{2}M + M_1 + m_2}
\]

### Step 4: Substitute \( a \) back to find \( T_1 \) and \( T_2 \)
Now, substitute \( a \) back into the expressions for \( T_1 \) and \( T_2 \):

\[
T_1 = m_1g - M_1 \left(\frac{m_1g - m_2g}{\frac{1}{2}M + M_1 + m_2}\right)
\]
\[
T_2 = m_2 \left(\frac{m_1g - m_2g}{\frac{1}{2}M + M_1 + m_2}\right) + m_2g
\]

### Final Expressions:
\[
a = \frac{m_1g - m_2g}{\frac{1}{2}M + M_1 + m_2}
\]
\[
T_1 = m_1g - M_1 \left(\frac{m_1g - m_2g}{\frac{1}{2}M + M_1 + m_2}\right)
\]
\[
T_2 = m_2 \left(\frac{m_1g - m_2g}{\frac{1}{2}M + M_1 + m_2}\right) + m_2g
\]

Would you like a detailed explanation of any part of the solution? 

Here are some related questions:
1. How would the solution change if \( m_1 = m_2 \)?
2. What happens to \( a \), \( T_1 \), and \( T_2 \) if \( M_1 \) becomes very large?
3. How does \( a \) behave as \( M \) increases?
4. What are the physical interpretations of \( T_1 \) and \( T_2 \) in this context?
5. Can you deduce a special case where \( T_1 = T_2 \)?

**Tip:** When solving for multiple variables, it's often useful to express one variable in terms of others and then substitute back into the remaining equations.

Learn how to solve the system of equations involving variables a, T1, and T2: M1a=m1g-T1, 1/2Ma=T1-T2, m2a=T2-m2g. This problem requires understanding linear algebra concepts and solving systems of linear equations. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation