To determine which box will have the greatest acceleration, we can use Newton's second law of motion, which states that:
\[ F = m \cdot a \]
where:
- \( F \) is the net force applied to the object,
- \( m \) is the mass of the object, and
- \( a \) is the acceleration of the object.
First, we need to calculate the net force acting on each box, considering that the gravitational force acting on each box will counteract part of the applied force.
The gravitational force acting on each box can be calculated with:
\[ F_{\text{gravity}} = m \cdot g \]
where \( g \) (acceleration due to gravity) is approximately 9.8 m/s².
Let's analyze each box:
-
20 kg box lifted with 40 N:
- Gravitational force: \( F_{\text{gravity}} = 20 , \text{kg} \cdot 9.8 , \text{m/s}^2 = 196 , \text{N} \)
- Net force: \( F_{\text{net}} = 40 , \text{N} - 196 , \text{N} = -156 , \text{N} \) (not lifting, acceleration is downward)
-
10 kg box lifted with 60 N:
- Gravitational force: \( F_{\text{gravity}} = 10 , \text{kg} \cdot 9.8 , \text{m/s}^2 = 98 , \text{N} \)
- Net force: \( F_{\text{net}} = 60 , \text{N} - 98 , \text{N} = -38 , \text{N} \) (not lifting, acceleration is downward)
-
10 kg box lifted with 40 N:
- Gravitational force: \( F_{\text{gravity}} = 10 , \text{kg} \cdot 9.8 , \text{m/s}^2 = 98 , \text{N} \)
- Net force: \( F_{\text{net}} = 40 , \text{N} - 98 , \text{N} = -58 , \text{N} \) (not lifting, acceleration is downward)
-
20 kg box lifted with 60 N:
- Gravitational force: \( F_{\text{gravity}} = 20 , \text{kg} \cdot 9.8 , \text{m/s}^2 = 196 , \text{N} \)
- Net force: \( F_{\text{net}} = 60 , \text{N} - 196 , \text{N} = -136 , \text{N} \) (not lifting, acceleration is downward)
Now, let's calculate the accelerations (and the sign indicates the direction, but we will focus on the absolute magnitude for comparison):
-
For the 10 kg box lifted with 60 N: \[ a = \frac{F_{\text{net}}}{m} = \frac{-38 , \text{N}}{10 , \text{kg}} = -3.8 , \text{m/s}^2 \]
-
For the 10 kg box lifted with 40 N: \[ a = \frac{F_{\text{net}}}{m} = \frac{-58 , \text{N}}{10 , \text{kg}} = -5.8 , \text{m/s}^2 \]
-
For the 20 kg box lifted with 60 N: \[ a = \frac{F_{\text{net}}}{m} = \frac{-136 , \text{N}}{20 , \text{kg}} = -6.8 , \text{m/s}^2 \]
-
For the 20 kg box lifted with 40 N: \[ a = \frac{F_{\text{net}}}{m} = \frac{-156 , \text{N}}{20 , \text{kg}} = -7.8 , \text{m/s}^2 \]
In terms of absolute value, the box with the greatest downward acceleration is the 10 kg box lifted with 60 N, yielding the least negative value of -3.8 m/s².
Thus, the box that will experience the greatest acceleration upward (the least downward) is the 10 kg box lifted with 60 N.