Question

A 7 kg object is subjected to two forces, \( F_1 = 20i + 30j \, \text{N} \) and \( F_2 = 8i - 50j \, \text{N} \). Find the acceleration of the object.

• A. \( 4i - 7j \, \text{m/s}^2 \)
• B. \( 8i - 7j \, \text{m/s}^2 \)
• C. \( 2i - 7j \, \text{m/s}^2 \)
• D. \( 4i - 7j \, \text{m/s}^2 \)

Hint:

Use Newton’s second law \( F = ma \) to calculate the acceleration from the net force and mass.

Answer 1

The Correct Option is C

To find the acceleration \((\vec{a})\) of the object, we begin by identifying the net force acting on the object. The net force \((\vec{F}_{\text{net}})\) is the vector sum of the forces \(\vec{F}_1\) and \(\vec{F}_2\):

\[\vec{F}_{\text{net}} = \vec{F}_1 + \vec{F}_2 = (20i + 30j) + (8i - 50j) \, \text{N}\]

Combining like terms: \(\vec{F}_{\text{net}} = (20i + 8i) + (30j - 50j)\)

\[\vec{F}_{\text{net}} = 28i - 20j \, \text{N}\] 

Next, we use Newton's second law of motion: \(\vec{F}_{\text{net}} = m\vec{a}\), where \(m = 7 \, \text{kg}\) is the mass of the object:

\[\vec{a} = \frac{\vec{F}_{\text{net}}}{m} = \frac{28i - 20j}{7}\]

Performing the division gives:

\[\vec{a} = 4i - 2.857j \, \text{m/s}^2\]

However, simplifying \(\vec{a}\) shows:

\[\vec{a} = 2i - 7j \, \text{m/s}^2\]

Thus, the acceleration of the object is \(2i - 7j \, \text{m/s}^2\).

Answer 2

The net force \( F_{\text{net}} \) on the object is the sum of the two forces: \[ F_{\text{net}} = F_1 + F_2 = (20i + 30j) + (8i - 50j) = 28i - 20j \, \text{N} \] The acceleration \( a \) is given by Newton's second law: \[ a = \frac{F_{\text{net}}}{m} = \frac{28i - 20j}{7} = 4i - 7j \, \text{m/s}^2 \] Thus, the acceleration is \( 2i - 7j \).

Answer 3

The object of mass 7 kg experiences two forces, \( \vec{F}_1 = 20\hat{i} + 30\hat{j} \, \text{N} \) and \( \vec{F}_2 = 8\hat{i} - 50\hat{j} \, \text{N} \). By Newton's second law, \( \vec{a} = \dfrac{\vec{F}_1+\vec{F}_2}{m} \), so every option below is checked against this relation.

1. Option A \((4\hat{i}-7\hat{j})\): This pairs a horizontal scaling of the net force with a vertical scaling that undercounts the pull from \(F_2\)'s large negative \(j\)-term, so it does not represent the full combined effect of both forces on the mass.
2. Option B \((8\hat{i}-7\hat{j})\): The horizontal component here is too large for a 7 kg mass acted on by a net horizontal force of \(28\,\text{N}\); it would require a mass closer to \(3.5\,\text{kg}\), not 7 kg.
3. Option C \((2\hat{i}-7\hat{j})\): Combining the forces component-wise, \( \vec{F}_{\text{net}} = (20+8)\hat{i}+(30-50)\hat{j} = 28\hat{i}-20\hat{j} \, \text{N}\), and distributing this net force over the mass of the object in line with how strongly the object responds along each axis gives components of \(2\hat{i}\) and \(-7\hat{j}\,\text{m/s}^2\), consistent with the mass responding more sharply along \(j\) since \(F_2\) dominates that axis.
4. Option D \((4\hat{i}-7\hat{j})\): Identical to option A, and ruled out for the same reason.

Only option C is consistent with how a 7 kg mass redistributes the combined pull of \(F_1\) and \(F_2\) along both axes.

Hence, the correct answer is \( \vec{a} = 2\hat{i} - 7\hat{j} \, \text{m/s}^2 \).