91Ó°ÊÓ

Calculating the net force acting on an object is an essential step in applying Newton's Second Law. To find the net force, combine all the individual forces acting on the object.
Forces are vector quantities, which means they have both a magnitude and a direction. When dealing with forces in problems like this, you'll often encounter them represented as vectors in a coordinate system.

• The net force (\( \vec{F}_\text{net} \)) is determined by vector addition of all acting forces.
• This ensures you take into account each force's direction while summing their magnitudes.

In the given problem, we are provided with two forces, \( \vec{F}_1 \) and \( \vec{F}_2 \), that act on a chopping block. By adding these two vectors together, we find the net force in each scenario.
For example, in Case (a), the forces \( \vec{F}_1 = (3.0 \mathrm{~N}) \hat{\mathrm{i}} + (4.0 \mathrm{~N}) \hat{\mathrm{j}} \) and \( \vec{F}_2 = (-3.0 \mathrm{~N}) \hat{\mathrm{i}} + (-4.0 \mathrm{~N}) \hat{\mathrm{j}} \) are added to find \( \vec{F}_\text{net} = (0 \mathrm{~N}) \hat{\mathrm{i}} + (0 \mathrm{~N}) \hat{\mathrm{j}} \).
Each case follows a similar process to ensure the net force is calculated accurately.

Understanding vector addition is crucial to solving problems involving forces in multiple directions. In most physical situations, forces do not always act neatly along one axis. Hence, you need to combine them to find the resultant force.
Vector addition involves two main components:

• Add vectors that are aligned along the same direction directly to get a total.
• Separately add their i-components (horizontal) and j-components (vertical) to find the net vector's magnitude and direction.

In our exercise, each force has two components: i-components along the x-axis and j-components along the y-axis. When dealing with vectors like \( \vec{F}_1 \) and \( \vec{F}_2 \), follow these steps:

• Align each vector's tail to the origin in the coordinate plane.
• Add up \( \hat{\mathrm{i}} \) components by summing their respective coefficients.
• Add up \( \hat{\mathrm{j}} \) components similarly.
• Resultant is the summation vector, which gives both direction and magnitude of your net force.

This approach helps create a clearer picture of how forces balance or cause motion in any direction.

After determining the net force, calculating the acceleration becomes straightforward using Newton's Second Law. This law articulates that the acceleration (\( \vec{a} \)) of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The formula used here is:\[ \vec{a} = \frac{\vec{F}_\text{net}}{m} \]Where:

• \( \vec{F}_\text{net} \) is the net force vector.
• \( m \) is the mass of the object (in kg).

In our problem, the chopping block has a mass of \( 2.0 \text{ kg} \). For each net force calculation in the exercise, substitute \( \vec{F}_\text{net} \) and the mass into the formula to find acceleration.
For instance, in Case (b), the net force is \( (0 \mathrm{~N}) \hat{\mathrm{i}} + (8.0 \mathrm{~N}) \hat{\mathrm{j}} \) and thus, the acceleration is calculated as:\[ \vec{a} = \frac{(0 \mathrm{~N}) \hat{\mathrm{i}} + (8.0 \mathrm{~N}) \hat{\mathrm{j}}}{2.0 \mathrm{~kg}} = (0 \mathrm{~m/s^2}) \hat{\mathrm{i}} + (4.0 \mathrm{~m/s^2}) \hat{\mathrm{j}} \]This method shows how forces result in acceleration, important especially in physics problems involving motion and dynamics.