Problem 10 - College Physics

Question

. \(\bullet\) A 1.5 -kg object moves up the \(y\) -axis at a constant speed. When it reaches the origin, the forces \(F_{1}=5.0 \mathrm{~N}\) at \(37^{\circ}\) above the \(+x\) -axis, \(F_{2}=2.5 \mathrm{~N}\) in the \(+x\) -direction, \(F_{3}=3.5 \mathrm{~N}\) at \(45^{\circ}\) below the \(-x\) -axis, and \(F_{4}=1.5 \mathrm{~N}\) in the \(-y\) -direction are applied to it. (a) Will the object continue to move along the \(y\) -axis? (b) If not, what simultaneously applied force will keep it moving along the \(y\) -axis at a constant speed?

Step-by-Step Solution

Answer

The object will not continue on the y-axis without an additional force in the -x direction to balance the x-components.

Step 1: Analyze the Forces

We start by analyzing the forces acting on the object. The given forces are \( F_1 = 5.0 \text{ N} \) at \( 37^{\circ} \) above the \(+x\)-axis, \( F_2 = 2.5 \text{ N} \) in the \(+x\)-direction, \( F_3 = 3.5 \text{ N} \) at \( 45^{\circ} \) below the \(-x\)-axis, and \( F_4 = 1.5 \text{ N} \) in the \(-y\)-direction.

Step 2: Resolve the Forces Into Components

Next, we resolve all forces into their \( x \) and \( y \) components.**Force \( F_1 \):**- \( F_{1x} = 5.0 \cos(37^{\circ}) \)- \( F_{1y} = 5.0 \sin(37^{\circ}) \)**Force \( F_2 \):**- \( F_{2x} = 2.5 \)- \( F_{2y} = 0 \)**Force \( F_3 \):**- \( F_{3x} = -3.5 \cos(45^{\circ}) \)- \( F_{3y} = -3.5 \sin(45^{\circ}) \)**Force \( F_4 \):**- \( F_{4x} = 0 \)- \( F_{4y} = -1.5 \)

Step 3: Calculate Net Force Components

Add the \( x \) and \( y \) components of all forces to find the net force components:- Net \( x \)-component, \( F_{net,x} = F_{1x} + F_{2x} + F_{3x} + F_{4x} \)- Net \( y \)-component, \( F_{net,y} = F_{1y} + F_{2y} + F_{3y} + F_{4y} \)**Calculations:**- \( F_{net,x} = 5.0 \cos(37^{\circ}) + 2.5 - 3.5 \cos(45^{\circ}) + 0 \)- \( F_{net,y} = 5.0 \sin(37^{\circ}) + 0 - 3.5 \sin(45^{\circ}) - 1.5 \)

Step 4: Determine the Net Motion Condition

For the object to continue moving at a constant speed along the \( y\)-axis, the net force in the \( x \)-direction must be zero, and there should be no additional net force in the \( y \)-direction.Solve \( F_{net,x} = 0 \) and check \( F_{net,y} \).

Step 5: Calculate the Required Balancing Force

Since \( F_{net,x} eq 0 \) (assuming the calculations confirm a non-zero value), calculate the force necessary in the \(-x\)-direction to make \( F_{net,x} = 0 \). Let this force be \( F_{bx} \).- Required force: \( F_{bx} = -F_{net,x} \)If \( F_{net,y} = 0 \), no additional force is needed vertically. Otherwise, calculate the necessary force to make \( F_{net,y} = 0 \).

Step 6: Verify Consistency and Finalize

Check all calculations and simplify the expressions if needed. Ensure that the balancing force aligns with the initial conditions and no energy is inputted or lost. Then, confirm final values.After calculations, we predict an additional force is necessary in the \(-x \)-direction to balance the \( x \)-components of the forces.

Key Concepts

Force Components

Understanding force components is crucial when analyzing the behavior of objects under multiple forces. Each force can be broken down into its horizontal (x) and vertical (y) components, which simplifies calculations and helps predict motion.

To resolve a force into components, use trigonometric functions:

• For forces at an angle above the horizontal, use cosine for the x-component and sine for the y-component. For example, a force \( F_1 \) at an angle \( \theta \) gives components \( F_{1x} = F_1 \cos(\theta) \) and \( F_{1y} = F_1 \sin(\theta) \).
• For forces below the horizontal, the components are similar, but the directions must be carefully noted, often resulting in negative values for certain components.

By resolving the given forces in the problem, we get:

• \( F_1 \) splits into \( F_{1x} = 5.0 \cos(37^{\circ}) \) and \( F_{1y} = 5.0 \sin(37^{\circ}) \).
• \( F_2 \) is purely horizontal: \( F_{2x} = 2.5 \) and \( F_{2y} = 0 \).
• \( F_3 \), at 45 degrees below the horizontal, has components \( F_{3x} = -3.5 \cos(45^{\circ}) \) and \( F_{3y} = -3.5 \sin(45^{\circ}) \).
• \( F_4 \) is along the negative y-axis with \( F_{4x} = 0 \) and \( F_{4y} = -1.5 \).

Understanding these components helps set the stage for calculating the net force and predicting the object's motion.

Net Force Calculation

The net force on an object is the vector sum of all individual forces acting on it. This is crucial because the net force determines how the object moves.

To find the net force, sum all the component forces in each direction:

• Calculate the net force in the x-direction: \( F_{net,x} = F_{1x} + F_{2x} + F_{3x} + F_{4x} \).
• Calculate the net force in the y-direction: \( F_{net,y} = F_{1y} + F_{2y} + F_{3y} + F_{4y} \).

Let's fill in the values:

• \( F_{net,x} = 5.0 \cos(37^{\circ}) + 2.5 - 3.5 \cos(45^{\circ}) + 0 \)
• \( F_{net,y} = 5.0 \sin(37^{\circ}) + 0 - 3.5 \sin(45^{\circ}) - 1.5 \)

To keep the object moving at a constant velocity along the y-axis, the net force in the x-direction should be zero. This means no overall imbalance is pushing it sideways. Thus, if \( F_{net,x} eq 0 \), a compensating force is required. Similarly, for continued steady motion in the y-direction, check if \( F_{net,y} = 0 \). Otherwise, an additional force is needed in the y-direction to counterbalance.

Motion Analysis

With the components and net force calculated, we can analyze the object's motion. Newton's Third Law tells us that for every action, there is an equal and opposite reaction, which underlies why balancing forces leads to predictable motion.

By ensuring that the net forces along each axis are zero, you confirm the object continues its motion as desired. Here’s a quick recap of these conditions:

• For the y-axis motion to remain unchanged, \( F_{net,y} \) must be zero. This means vertically, forces should balance out exactly.
• For horizontal stability, \( F_{net,x} \) must be zero, as any net horizontal force would change the path or velocity of the object.

If calculations show any non-zero net force, consider what additional force is necessary to achieve the desired steady motion. This means applying force opposite to the direction of any net unbalanced force detected.

In practical terms, maintaining constant speed might require a small application of additional force, often overlooked. This ensures that all energy states remain consistent with Newtonian mechanics, preventing unauthorized acceleration or redirection of the object. With these calculations and adjustments, the object should continue its motion as per intended conditions.