Problem 72 - Master Resource Book in JEE Main Physics

Question

Following forces start acting on a particle at rest at the origin of the coordinate system simultaneously \(\mathbf{F}_{1}=5 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}, \mathbf{F}_{2}=2 \hat{\mathbf{i}}+8 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}, \mathbf{F}_{3}=-6 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-7 \hat{\mathbf{k}}\) \(\mathbf{F}_{4}=-\hat{\mathbf{i}}-3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}\). The particle will move (a) in \(x-y\) plane (b) in \(y-z\) plane (c) in \(x-z\) plane (d) along \(x\)-axis

Step-by-Step Solution

Answer

The particle will move in the y-z plane.

Step 1: Sum Forces in each direction

First, we sum up all the forces in the x, y, and z directions separately. For the x-component, we have:\[ F_{x} = 5 + 2 - 6 - 1 = 0 \]For the y-component, we have:\[ F_{y} = -5 + 8 + 4 - 3 = 4 \]For the z-component, we have:\[ F_{z} = 5 + 6 - 7 - 2 = 2 \]

Step 2: Determine the plane of motion

Since the net force in the x-direction is zero, and there are non-zero forces in the y and z directions, the particle will move in the plane spanned by these forces.

Step 3: Planes Available

The options for movement are in the x-y plane, y-z plane, x-z plane, or along the x-axis (which would mean all other forces result in a cancelled effect). Since the net force is only on the y and z axes, the motion will be confined to the y-z plane.

Key Concepts

Vector Addition

Understanding vector addition is essential when dealing with forces in three-dimensional space.
A vector is represented by both a magnitude and a direction, and we can add vectors together to find the resultant or total effect of such forces.
When forces act on an object, it's important to sum up all these vectors to see the net effect.

• Vectors are usually represented using unit vectors, like \( \hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}} \), which correspond to the x, y, and z directions respectively.
• To add vectors, you combine them component-wise: combine all x-components, all y-components, and all z-components from each vector.
• In our example, multiple forces act on a particle: \( \mathbf{F}_{1}, \mathbf{F}_{2}, \mathbf{F}_{3}, \mathbf{F}_{4} \).
• The x-components, y-components, and z-components are added separately, yielding \( F_x = 0 \), \( F_y = 4 \), and \( F_z = 2 \).

Vector addition helps simplify complex situations by allowing us to find one net force that represents the combined effect of multiple forces.

Net Force Calculation

The net force is the vector sum of all forces acting upon a particle.
Calculating it involves combining all individual force components to determine the overall force direction and magnitude.
When analyzing an object's motion, understanding net force is crucial because it dictates the direction and speed of the particle.

• The net force algorithm assumes linear vector addition which combines each force component.
• In the given problem, the forces are defined as \( \mathbf{F}_{1}, \mathbf{F}_{2}, \mathbf{F}_{3}, \mathbf{F}_{4} \).
• Upon calculating the components: \( F_x = 0 \), \( F_y = 4 \), \( F_z = 2 \).
• This calculation implies that while the net force along the x-axis is zero, the particle experiences a force in the y and z directions.

As a result, the net force not only determines motion but also provides directionality information, essential for further position and velocity calculations.

Plane of Motion

The plane of motion is determined by identifying the directions where net forces exist.
In essence, it refers to choosing the plane in which a particle will move based on its net force components.

• Considering the net forces, you need to look at non-zero components to establish possible planes of movement.
• In the example provided, the calculations show \( F_x = 0 \), while \( F_y = 4 \) and \( F_z = 2 \) are non-zero.
• Given that the x-component is zero, the particle moves in the y-z plane because those are the axes along which net forces act.
• This aligns with physical intuition — no force in one direction means no movement in that direction.

Understanding the plane of motion is key in predicting the trajectory and understanding the movement's constraints.