At an instant when a 4.0-kg object has an acceleration equal to (5i + 3j) m/s2, one of the two forces acting on the object is known to be (12i + 22j) N. Determine the magnitude of the other force acting on the object.
1) 2.0 N
2) 13 N
3) 18 N
4) 1.7 N
5) 20 N
The Correct Answer and Explanation is:
To solve this problem, we need to apply Newton’s Second Law of Motion, which states:
[
\mathbf{F}_{\text{net}} = m \cdot \mathbf{a}
]
Where:
- (\mathbf{F}_{\text{net}}) is the net force acting on the object.
- (m) is the mass of the object.
- (\mathbf{a}) is the acceleration vector of the object.
Step 1: Find the net force vector
Given that the mass of the object is ( m = 4.0 \, \text{kg} ) and the acceleration is (\mathbf{a} = (5i + 3j) \, \text{m/s}^2), we can use Newton’s second law to find the net force:
[
\mathbf{F}_{\text{net}} = m \cdot \mathbf{a} = (4.0 \, \text{kg}) \cdot (5i + 3j) \, \text{m/s}^2
]
[
\mathbf{F}_{\text{net}} = (20i + 12j) \, \text{N}
]
So, the net force acting on the object is:
[
\mathbf{F}_{\text{net}} = 20 \hat{i} + 12 \hat{j} \, \text{N}
]
Step 2: Break the problem into two forces
We know that there are two forces acting on the object. One of them is (\mathbf{F}1 = (12i + 22j) \, \text{N}), and the net force is (\mathbf{F}{\text{net}} = 20i + 12j \, \text{N}).
Using the principle of superposition, the net force is the vector sum of the two forces:
[
\mathbf{F}_{\text{net}} = \mathbf{F}_1 + \mathbf{F}_2
]
We can solve for the second force (\mathbf{F}_2):
[
\mathbf{F}2 = \mathbf{F}{\text{net}} – \mathbf{F}_1
]
Substitute the values of (\mathbf{F}_{\text{net}}) and (\mathbf{F}_1):
[
\mathbf{F}_2 = (20i + 12j) – (12i + 22j)
]
[
\mathbf{F}_2 = (20i – 12i) + (12j – 22j)
]
[
\mathbf{F}_2 = 8i – 10j \, \text{N}
]
Step 3: Find the magnitude of (\mathbf{F}_2)
To find the magnitude of (\mathbf{F}_2), we use the Pythagorean theorem:
[
|\mathbf{F}_2| = \sqrt{(8)^2 + (-10)^2}
]
[
|\mathbf{F}_2| = \sqrt{64 + 100} = \sqrt{164}
]
[
|\mathbf{F}_2| \approx 12.8 \, \text{N}
]
The closest answer choice is 13 N, so the magnitude of the other force is approximately 13 N.
Conclusion
The correct answer is 2) 13 N.