Problem 59 - University Physics with Modern Physics

Question

CALC A radio-controlled model airplane has a momentum given by \(\left[\left(-0.75 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}^{3}\right) t^{2}+(3.0 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s})\right] \hat{\imath}+(0.25 \mathrm{kg} \cdot \) \(\mathrm{m} / \mathrm{s}^{2} ) t \hat{J} .\) What are the \(x-, y-\) and \(z\) -components of the net force on the airplane?

Step-by-Step Solution

Answer

The net force components are: \( F_x = -1.5 \, \text{N} \cdot t \), \( F_y = 0.25 \, \text{N} \), \( F_z = 0 \, \text{N} \).

Step 1: Identify the Given Momentum

The momentum vector of the airplane is given as \( \left[ (-0.75 \, \text{kg} \cdot \text{m/s}^3) \cdot t^2 + (3.0 \, \text{kg} \cdot \text{m/s}) \right] \hat{\imath} + (0.25 \, \text{kg} \cdot \text{m/s}^2) \cdot t \hat{\jmath} \). We are tasked with finding the components of the net force, which involves taking derivatives since force is the rate of change of momentum.

Step 2: Calculate the Net Force in the x-direction

The momentum in the x-direction, \( p_x \), is \( (-0.75 \, \text{kg} \cdot \text{m/s}^3) \cdot t^2 + (3.0 \, \text{kg} \cdot \text{m/s}) \). To find the force \( F_x \), take the derivative of \( p_x \) with respect to time \( t \): \[ F_x = \frac{d}{dt} \left[ (-0.75 \, t^2 + 3.0) \right] = -1.5 \, \cdot t \]

Key Concepts

Momentum

Momentum is a physical quantity that represents the motion of an object. It's calculated by multiplying the mass of the object by its velocity. In calculus-based physics, momentum is a vector quantity, which means it has both magnitude and direction. The equation for momentum is given as \( \vec{p} = m \vec{v} \), where \( m \) denotes mass and \( \vec{v} \) denotes velocity. This equation shows how both the speed and direction of an object affect momentum.

In the exercise, the momentum is expressed as a function of time, which implies it can change as time progresses. The given momentum vector incorporates time-dependent components that allow us to calculate the changing force acting on the object.

Net Force

Net force is the total force acting on an object, considering all individual forces. According to Newton's Second Law of Motion, net force can change an object's momentum. The law is formulated as \( \vec{F} = \frac{d \vec{p}}{dt} \), meaning that net force is equivalent to the time derivative of the momentum.

In situations where forces vary with time, net force can be derived by taking the derivative of the momentum function with respect to time. This approach helps us analyze how quickly momentum changes, providing us with the force applied to the object at any moment.

Derivatives

Derivatives are mathematical tools derived from calculus that show how a function changes at any given point. They represent the rate of change of a quantity. In our context, the derivative of the momentum with respect to time gives us the force acting on the object. This relationship stems from Newton's Second Law.

For the given problem, the x-component of the momentum was differentiated to find the rate of change, which corresponds to the force in the x-direction. Initially, the momentum was \( (-0.75 \, \text{kg} \cdot \text{m/s}^3) \cdot t^2 + (3.0 \, \text{kg} \cdot \text{m/s}) \), and upon taking its derivative with an emphasis on the time variable \( t \), the force in the x-direction \( F_x \) was determined as \( -1.5 \, \cdot t \). This calculation demonstrates how derivatives are used to obtain forces.

Vector Components

Vector components are crucial in physics as they help decompose vectors into orthogonal parts, generally along the \( x \), \( y \), and \( z \) axes in three-dimensional space. Every vector can be broken into these components based on its direction in space.

In this exercise, the original problem provided a momentum vector function featuring separate terms for both the \( \hat{\imath} \) (x-direction) and \( \hat{\jmath} \) (y-direction) components. Each component represents a different dimension of the vector's influence. By taking derivatives of each component separately, one can find separate forces acting in the x and y directions. This is a common approach in vector analysis, helping isolate the effects acting along each axis to analyze motion more effectively.