Problem 56 - University Physics with Modern Physics

Question

An object of mass \(m\) is at rest in equilibrium at the origin. At \(t =\) 0 a new force \(\vec{F}(t)\) is applied that has components $$F_x(t) = k_1 + k_2y$$ $$F_y(t) = k_3t$$ where \(k_1\), \(k_2\), and \(k_3\) are constants. Calculate the position \(\vec{r}(t)\) and velocity \(\vec{v}(t)\) vectors as functions of time.

Step-by-Step Solution

Answer

\( x(t) = \frac{k_1}{2m} t^2 + \frac{k_2k_3}{36m^2} t^6, \; y(t) = \frac{k_3}{6m} t^3 \). \( v_x(t) = \frac{k_1}{m}t, \; v_y(t) = \frac{k_3}{2m} t^2 \).

Step 1: Understanding the Forces

The force acting on the object has two components: \( F_x(t) = k_1 + k_2y \) and \( F_y(t) = k_3t \). These forces will influence the motion of the object along the x and y axes.

Step 2: Apply Newton's Second Law

According to Newton's second law, \( \vec{F} = m\vec{a} \), where \( \vec{a} \) is the acceleration. Therefore, \( F_x = ma_x = k_1 + k_2y \) and \( F_y = ma_y = k_3t \).

Step 3: Solve for Acceleration

Rearrange Newton's second law to find accelerations:- For x-component: \( a_x = \frac{k_1 + k_2y}{m} \)- For y-component: \( a_y = \frac{k_3t}{m} \)

Step 4: Integrate to Find Velocity

Integrate the acceleration with respect to time to find velocity:- In x-direction: \( v_x(t) = \int a_x \, dt = \frac{k_1 + k_2y}{m}t + C_x \)- In y-direction: \( v_y(t) = \int a_y \, dt = \frac{k_3}{2m} t^2 + C_y \)Assuming initial velocities to be zero: \( C_x = 0 \) and \( C_y = 0 \).

Step 5: Solve Dynamic Equation for y

Since \( a_x \) depends on \( y \), use the kinematic relation for \( a_y \) to find \( y \):Given \( a_y = \frac{k_3t}{m} \), integrate once more:\( y(t) = \int v_y(t) \, dt = \frac{k_3}{6m} t^3 \).

Step 6: Integrate Again for Position

Integrate velocities to find positions: - For the x-component: \( x(t) = \int v_x(t) \, dt = \frac{k_1}{2m}t^2 + \frac{k_2}{6m}y t^3 \).- For the y-component: \( y(t) = \frac{k_3}{6m} t^3 \). Adjust according to calculated dynamic equation solutions.

Step 7: Simplify and Substitute to Find \( x(t) \)

Substitute \( y(t) = \frac{k_3}{6m} t^3 \) into \( x(t) \): \( x(t) = \frac{k_1}{2m}t^2 + \frac{k_2}{6m}\left(\frac{k_3}{6m} t^3\right) t^3 \).This simplifies to \( x(t) = \frac{k_1}{2m}t^2 + \frac{k_2k_3}{36m^2} t^6 \).

Key Concepts

Dynamics

Dynamics is the study of motion and its causes. It helps us understand why objects move the way they do. In this exercise, we focus on an object that switches from rest to motion because of forces acting upon it. Dynamics is crucial here because it connects the dots between forces, motion, and time.
In this problem, a new force is introduced at time zero, causing the object to accelerate from rest. The study of dynamics here revolves around how these forces, with their unique components, affect the object's motion along different directions. It's like putting together a puzzle to see the full picture of the object's trajectory.

Force Components

Force components are the parts of a force that act in different directions. In this exercise, the force \( \vec{F}(t) \) acting upon the object is split into two components: \( F_x(t) = k_1 + k_2y \) and \( F_y(t) = k_3t \). These components show us how the force affects movement in the x and y directions separately.
Force components break down the applied force into fundamental pieces, making it easier to analyze the movement caused along each axis. By knowing how each component acts on its respective axis, we can simplify complex motion dynamics into more manageable calculations. This separation becomes very insightful for predicting how the object will travel along its path as time progresses.

Equilibrium

Equilibrium is a state where all forces acting on an object are balanced, resulting in no net force. An object in equilibrium remains at rest or moves at a constant velocity. In this problem, the object initially is at rest at the origin, meaning it is in equilibrium.
However, once \( \vec{F}(t) \) is applied, the equilibrium is disrupted, giving rise to movement. Understanding equilibrium helps us recognize at what point the object transitions from stillness to motion, providing a baseline to determine how external forces alter its state. Tracking these changes lets us understand the shift in dynamics.

Acceleration

Acceleration is the rate at which an object's velocity changes with time. It is a core element of Newton's Second Law, which relates acceleration to force and mass with \( \vec{F} = m\vec{a} \).
In this problem, we calculate the object's acceleration in both the x and y directions using the given force components. By rearranging Newton's Second Law for each direction, we arrive at:

• For x-direction: \( a_x = \frac{k_1 + k_2y}{m} \)
• For y-direction: \( a_y = \frac{k_3t}{m} \)

Integrating these acceleration components gives us the velocity functions of time, explaining how quickly the object's speed changes in each direction. Understanding acceleration helps track how the object speeds up as it's continuously pushed by varying forces.