Newton's second law for translation motion in the xy plane is `sumvec(F) = mvec(a) `, Newton's second law for rotation is `sumT_(x) = lalpha_(x) `. Consider the case of a particle moving in the x-y plane under the influence of a single force.

Options

A. Both `vec(F) = m vec(a) ` and `T_(x) = I alpha_(x)` must be used to analyze the motion of this particle .
B. Either `vec( F ) = m vec( a) ` or `T_(x) -lalpha_(x)` , can be used to analyze the motion of this particle.
C. Only `vec( F) = m vec( a)` needs to be used to analyze the motoin of this particle
D. Only `T_(x) = lalpha_(x)` can be used to analyze the motion of this particle.

Answer

To solve the problem regarding the motion of a particle in the xy-plane under the influence of a single force, we will analyze both translational and rotational motion using Newton's second laws.

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Particle**:
- We start by recognizing that the particle is subjected to a single force, which we can denote as \( \vec{F} \). This force will influence the particle's motion in both the x and y directions.

2. **Apply Newton's Second Law for Translation**:
- According to Newton's second law for translational motion in the xy-plane, we have:
\[ \sum \vec{F} = m \vec{a} \]
- Here, \( \sum \vec{F} \) represents the vector sum of all forces acting on the particle, \( m \) is the mass of the particle, and \( \vec{a} \) is the acceleration vector of the particle.

3. **Resolve the Force into Components**:
- If the force \( \vec{F} \) has components \( F_x \) and \( F_y \), we can express it as:
\[ \vec{F} = F_x \hat{i} + F_y \hat{j} \]
- The acceleration can also be resolved into its components:
\[ \vec{a} = a_x \hat{i} + a_y \hat{j} \]

4. **Set Up the Equations for Each Direction**:
- From the translational motion equation, we can write two separate equations for the x and y directions:
\[ F_x = m a_x \quad \text{(1)} \]
\[ F_y = m a_y \quad \text{(2)} \]

5. **Apply Newton's Second Law for Rotation**:
- If the force \( \vec{F} \) acts at a distance \( r \) from the axis of rotation, we can consider the torque \( \tau \) produced by this force:
\[ \tau = r \times F \]
- According to Newton's second law for rotation, we have:
\[ \sum \tau = I \alpha \]
- Here, \( I \) is the moment of inertia about the axis of rotation, and \( \alpha \) is the angular acceleration.

6. **Combine Both Analyses**:
- For a complete analysis of the motion of the particle, we must consider both translational and rotational equations. Therefore, we will use both:
\[ \sum \vec{F} = m \vec{a} \quad \text{and} \quad \sum \tau = I \alpha \]

### Conclusion:
To analyze the motion of a particle in the xy-plane under the influence of a single force, we utilize both translational and rotational forms of Newton's second law. Thus, the correct approach is to apply both \( \sum \vec{F} = m \vec{a} \) and \( \sum \tau = I \alpha \).