Answer :
Alright, let’s solve this step by step for each case.
### Case 1: [tex]\( M = 13 \text{ kg}, m = 6 \text{ kg}, a = 11.4 \text{ m/s}^2 \)[/tex]
Objective: Calculate the tensions [tex]\(T_1\)[/tex] and [tex]\(T_2\)[/tex].
1. Determine the tension [tex]\( T_1 \)[/tex] in the string on the [tex]\( M \)[/tex] side:
[tex]\[ T_1 = M \cdot g + M \cdot a \][/tex]
Given:
[tex]\[ M = 13 \text{ kg} \][/tex]
[tex]\[ g = 9.8 \text{ m/s}^2 \][/tex]
[tex]\[ a = 11.4 \text{ m/s}^2 \][/tex]
[tex]\[ T_1 = 13 \cdot 9.8 + 13 \cdot 11.4 \][/tex]
[tex]\[ T_1 = 127.4 + 148.2 \][/tex]
[tex]\[ T_1 = 275.6 \text{ N} \][/tex]
2. Determine the tension [tex]\( T_2 \)[/tex] in the string on the [tex]\( m \)[/tex] side:
[tex]\[ T_2 = m \cdot g - m \cdot a \][/tex]
Given:
[tex]\[ m = 6 \text{ kg} \][/tex]
[tex]\[ g = 9.8 \text{ m/s}^2 \][/tex]
[tex]\[ a = 11.4 \text{ m/s}^2 \][/tex]
[tex]\[ T_2 = 6 \cdot 9.8 - 6 \cdot 11.4 \][/tex]
[tex]\[ T_2 = 58.8 - 68.4 \][/tex]
[tex]\[ T_2 = -9.6 \text{ N} \][/tex]
### Case 2: [tex]\( M = 13 \text{ kg}, m = 9 \text{ kg}, a = 1.4 \text{ m/s}^2 \)[/tex]
Objective: Calculate the tensions [tex]\(T_1\)[/tex] and [tex]\(T_2\)[/tex].
1. Determine the tension [tex]\( T_1 \)[/tex] in the string on the [tex]\( M \)[/tex] side:
[tex]\[ T_1 = M \cdot g + M \cdot a \][/tex]
Given:
[tex]\[ M = 13 \text{ kg} \][/tex]
[tex]\[ g = 9.8 \text{ m/s}^2 \][/tex]
[tex]\[ a = 1.4 \text{ m/s}^2 \][/tex]
[tex]\[ T_1 = 13 \cdot 9.8 + 13 \cdot 1.4 \][/tex]
[tex]\[ T_1 = 127.4 + 18.2 \][/tex]
[tex]\[ T_1 = 145.6 \text{ N} \][/tex]
2. Determine the tension [tex]\( T_2 \)[/tex] in the string on the [tex]\( m \)[/tex] side:
[tex]\[ T_2 = m \cdot g - m \cdot a \][/tex]
Given:
[tex]\[ m = 9 \text{ kg} \][/tex]
[tex]\[ g = 9.8 \text{ m/s}^2 \][/tex]
[tex]\[ a = 1.4 \text{ m/s}^2 \][/tex]
[tex]\[ T_2 = 9 \cdot 9.8 - 9 \cdot 1.4 \][/tex]
[tex]\[ T_2 = 88.2 - 12.6 \][/tex]
[tex]\[ T_2 = 75.6 \text{ N} \][/tex]
### Case 3: [tex]\( M = 13 \text{ kg}, m = 12 \text{ kg}, a = ? \)[/tex]
Objective: Calculate the acceleration [tex]\(a\)[/tex].
The acceleration [tex]\( a \)[/tex] can be calculated using the following formula:
[tex]\[ a = \frac{(M - m) \cdot g}{M + m} \][/tex]
Given:
[tex]\[ M = 13 \text{ kg} \][/tex]
[tex]\[ m = 12 \text{ kg} \][/tex]
[tex]\[ g = 9.8 \text{ m/s}^2 \][/tex]
[tex]\[ a = \frac{(13 - 12) \cdot 9.8}{13 + 12} \][/tex]
[tex]\[ a = \frac{1 \cdot 9.8}{25} \][/tex]
[tex]\[ a = \frac{9.8}{25} \][/tex]
[tex]\[ a = 0.392 \text{ m/s}^2 \][/tex]
### Case 4: [tex]\( M = 16 \text{ kg}, m = 12 \text{ kg}, a = ? \)[/tex]
Objective: Calculate the acceleration [tex]\(a\)[/tex].
The acceleration [tex]\( a \)[/tex] can be calculated using the following formula:
[tex]\[ a = \frac{(M - m) \cdot g}{M + m} \][/tex]
Given:
[tex]\[ M = 16 \text{ kg} \][/tex]
[tex]\[ m = 12 \text{ kg} \][/tex]
[tex]\[ g = 9.8 \text{ m/s}^2 \][/tex]
[tex]\[ a = \frac{(16 - 12) \cdot 9.8}{16 + 12} \][/tex]
[tex]\[ a = \frac{4 \cdot 9.8}{28} \][/tex]
[tex]\[ a = \frac{39.2}{28} \][/tex]
[tex]\[ a = 1.4 \text{ m/s}^2 \][/tex]
### Summary of Results:
- Case 1: [tex]\( T_1 = 275.6 \, \text{N}, \, T_2 = -9.6 \, \text{N} \)[/tex]
- Case 2: [tex]\( T_1 = 145.6 \, \text{N}, \, T_2 = 75.6 \, \text{N} \)[/tex]
- Case 3: [tex]\( a = 0.392 \, \text{m/s}^2 \)[/tex]
- Case 4: [tex]\( a = 1.4 \, \text{m/s}^2 \)[/tex]
These calculations are directly verifying the dynamics and tension in the system under the given conditions.
### Case 1: [tex]\( M = 13 \text{ kg}, m = 6 \text{ kg}, a = 11.4 \text{ m/s}^2 \)[/tex]
Objective: Calculate the tensions [tex]\(T_1\)[/tex] and [tex]\(T_2\)[/tex].
1. Determine the tension [tex]\( T_1 \)[/tex] in the string on the [tex]\( M \)[/tex] side:
[tex]\[ T_1 = M \cdot g + M \cdot a \][/tex]
Given:
[tex]\[ M = 13 \text{ kg} \][/tex]
[tex]\[ g = 9.8 \text{ m/s}^2 \][/tex]
[tex]\[ a = 11.4 \text{ m/s}^2 \][/tex]
[tex]\[ T_1 = 13 \cdot 9.8 + 13 \cdot 11.4 \][/tex]
[tex]\[ T_1 = 127.4 + 148.2 \][/tex]
[tex]\[ T_1 = 275.6 \text{ N} \][/tex]
2. Determine the tension [tex]\( T_2 \)[/tex] in the string on the [tex]\( m \)[/tex] side:
[tex]\[ T_2 = m \cdot g - m \cdot a \][/tex]
Given:
[tex]\[ m = 6 \text{ kg} \][/tex]
[tex]\[ g = 9.8 \text{ m/s}^2 \][/tex]
[tex]\[ a = 11.4 \text{ m/s}^2 \][/tex]
[tex]\[ T_2 = 6 \cdot 9.8 - 6 \cdot 11.4 \][/tex]
[tex]\[ T_2 = 58.8 - 68.4 \][/tex]
[tex]\[ T_2 = -9.6 \text{ N} \][/tex]
### Case 2: [tex]\( M = 13 \text{ kg}, m = 9 \text{ kg}, a = 1.4 \text{ m/s}^2 \)[/tex]
Objective: Calculate the tensions [tex]\(T_1\)[/tex] and [tex]\(T_2\)[/tex].
1. Determine the tension [tex]\( T_1 \)[/tex] in the string on the [tex]\( M \)[/tex] side:
[tex]\[ T_1 = M \cdot g + M \cdot a \][/tex]
Given:
[tex]\[ M = 13 \text{ kg} \][/tex]
[tex]\[ g = 9.8 \text{ m/s}^2 \][/tex]
[tex]\[ a = 1.4 \text{ m/s}^2 \][/tex]
[tex]\[ T_1 = 13 \cdot 9.8 + 13 \cdot 1.4 \][/tex]
[tex]\[ T_1 = 127.4 + 18.2 \][/tex]
[tex]\[ T_1 = 145.6 \text{ N} \][/tex]
2. Determine the tension [tex]\( T_2 \)[/tex] in the string on the [tex]\( m \)[/tex] side:
[tex]\[ T_2 = m \cdot g - m \cdot a \][/tex]
Given:
[tex]\[ m = 9 \text{ kg} \][/tex]
[tex]\[ g = 9.8 \text{ m/s}^2 \][/tex]
[tex]\[ a = 1.4 \text{ m/s}^2 \][/tex]
[tex]\[ T_2 = 9 \cdot 9.8 - 9 \cdot 1.4 \][/tex]
[tex]\[ T_2 = 88.2 - 12.6 \][/tex]
[tex]\[ T_2 = 75.6 \text{ N} \][/tex]
### Case 3: [tex]\( M = 13 \text{ kg}, m = 12 \text{ kg}, a = ? \)[/tex]
Objective: Calculate the acceleration [tex]\(a\)[/tex].
The acceleration [tex]\( a \)[/tex] can be calculated using the following formula:
[tex]\[ a = \frac{(M - m) \cdot g}{M + m} \][/tex]
Given:
[tex]\[ M = 13 \text{ kg} \][/tex]
[tex]\[ m = 12 \text{ kg} \][/tex]
[tex]\[ g = 9.8 \text{ m/s}^2 \][/tex]
[tex]\[ a = \frac{(13 - 12) \cdot 9.8}{13 + 12} \][/tex]
[tex]\[ a = \frac{1 \cdot 9.8}{25} \][/tex]
[tex]\[ a = \frac{9.8}{25} \][/tex]
[tex]\[ a = 0.392 \text{ m/s}^2 \][/tex]
### Case 4: [tex]\( M = 16 \text{ kg}, m = 12 \text{ kg}, a = ? \)[/tex]
Objective: Calculate the acceleration [tex]\(a\)[/tex].
The acceleration [tex]\( a \)[/tex] can be calculated using the following formula:
[tex]\[ a = \frac{(M - m) \cdot g}{M + m} \][/tex]
Given:
[tex]\[ M = 16 \text{ kg} \][/tex]
[tex]\[ m = 12 \text{ kg} \][/tex]
[tex]\[ g = 9.8 \text{ m/s}^2 \][/tex]
[tex]\[ a = \frac{(16 - 12) \cdot 9.8}{16 + 12} \][/tex]
[tex]\[ a = \frac{4 \cdot 9.8}{28} \][/tex]
[tex]\[ a = \frac{39.2}{28} \][/tex]
[tex]\[ a = 1.4 \text{ m/s}^2 \][/tex]
### Summary of Results:
- Case 1: [tex]\( T_1 = 275.6 \, \text{N}, \, T_2 = -9.6 \, \text{N} \)[/tex]
- Case 2: [tex]\( T_1 = 145.6 \, \text{N}, \, T_2 = 75.6 \, \text{N} \)[/tex]
- Case 3: [tex]\( a = 0.392 \, \text{m/s}^2 \)[/tex]
- Case 4: [tex]\( a = 1.4 \, \text{m/s}^2 \)[/tex]
These calculations are directly verifying the dynamics and tension in the system under the given conditions.
