Problem 16 - University Physics with Modern Physics

Question

CALC Starting at \(t=0,\) a horizontal net force \(\vec{F}=\) \((0.280 \mathrm{N} / \mathrm{s}) \hat{t} \hat{\imath}+\left(-0.450 \mathrm{N} / \mathrm{s}^{2}\right) t^{2} \hat{\mathrm{J}}\) is applied to a box that has an initial momentum \(\vec{p}=(-3.00 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}) \hat{\imath}+(4.00 \mathrm{kg} \cdot \) \(\mathrm{m} / \mathrm{s} ) \hat{\boldsymbol{J}}\) . What is the momentum of the box at \(t=2.00 \mathrm{s} ?\)

Step-by-Step Solution

Answer

The momentum of the box at \(t=2\,\text{s}\) is \((-2.44\,\text{kg}\cdot\text{m/s}) \hat{\imath} + (2.80\,\text{kg}\cdot\text{m/s}) \hat{\jmath}\).

Step 1: Understand the Net Force

The net force applied to the box is given as \( \vec{F}(t) = (0.280\, \text{N/s})t \hat{\imath} + (-0.450\, \text{N/s}^2)t^2 \hat{\jmath} \). It varies with time, so we need to integrate it over time to find the change in momentum.

Step 2: Use Impulse-Momentum Theorem

According to the impulse-momentum theorem, the change in momentum \( \Delta \vec{p} \) is equal to the integral of force with respect to time, i.e., \( \Delta \vec{p} = \int_{0}^{2} \vec{F}(t) \, dt \).

Step 3: Set Up the Integral for \( \hat{\imath} \) Component

For the \( \hat{\imath} \) component: \( \Delta p_{i} = \int_{0}^{2} (0.280t) \, dt \). This simplifies to \( 0.280 \int_{0}^{2} t \, dt \).

Step 4: Calculate the \( \hat{\imath} \) Component Change in Momentum

Solve the integral: \[ \Delta p_{i} = 0.280 \left[ \frac{t^2}{2} \right]_{0}^{2} = 0.280 \left( \frac{4}{2} - 0 \right) = 0.280 \times 2 = 0.560 \text{ kg} \cdot \text{m/s}. \]

Step 5: Set Up the Integral for \( \hat{\jmath} \) Component

For the \( \hat{\jmath} \) component: \( \Delta p_{j} = \int_{0}^{2} (-0.450 t^2) \, dt \). This simplifies to \( -0.450 \int_{0}^{2} t^2 \, dt \).

Step 6: Calculate the \( \hat{\jmath} \) Component Change in Momentum

Solve the integral: \[ \Delta p_{j} = -0.450 \left[ \frac{t^3}{3} \right]_{0}^{2} = -0.450 \left( \frac{8}{3} \right) = -0.450 \times \frac{8}{3} = -1.200 \text{ kg} \cdot \text{m/s}. \]

Step 7: Determine Final Momentum

The initial momentum is given by \( \vec{p}_{initial} = (-3.00\, \text{kg}\cdot\text{m/s}) \hat{\imath} + (4.00\, \text{kg}\cdot\text{m/s}) \hat{\jmath} \). Add the change in momentum to the initial momentum: \[ \vec{p}_{final} = \vec{p}_{initial} + \Delta \vec{p} = (-3.00 + 0.560) \hat{\imath} + (4.00 - 1.200) \hat{\jmath}. \]

Step 8: Calculate the Final Momentum

Calculate each component: \( \hat{\imath} \) component: \(-3.00 + 0.560 = -2.44\, \text{kg}\cdot\text{m/s} \) \( \hat{\jmath} \) component: \(4.00 - 1.200 = 2.80\, \text{kg}\cdot\text{m/s} \) Thus, \( \vec{p}_{final} = (-2.44\, \text{kg}\cdot\text{m/s}) \hat{\imath} + (2.80\, \text{kg}\cdot\text{m/s}) \hat{\jmath}. \)

Key Concepts

Impulse-Momentum Theorem

The impulse-momentum theorem tells us an important fact about force and motion. According to this theorem, the change in an object's momentum is equal to the impulse applied to it. Impulse, in turn, is the integral of the net force over time.
This means that if you apply a force to an object for a certain amount of time, you change its momentum based on the size of the force and how long it's applied. Mathematically, \[ \Delta \vec{p} = \int \vec{F}(t) \, dt \]
In our exercise, the net force is time-dependent, so we integrate it over our desired time frame to find out how much the momentum has changed.

• Impulse represents the area under a force-time graph.
• Momentum depends on both the mass and velocity of an object, so it's a vector quantity.

This theorem is crucial for connecting force and motion parameters in dynamic systems.

Integral Calculus in Physics

Integral calculus is fundamental in physics to calculate quantities that accumulate over time, like displacement or change in momentum. In the impulse-momentum theorem, we use integral calculus to compute the impulse.
We come across integrals that represent the area under a curve on a graph, which in this context relates to force versus time or acceleration.
For our exercise, we need to integrate the given force function over time from 0 to 2 seconds to find changes in the different components of momentum.

• We compute the definite integrals for each component of force - the results give the changes in the respective components of momentum.
• The integral helps us find accumulated quantities over time, making it ideal for dynamic analyses.
• In vector calculus, each component (such as \(\hat{\imath}\) or \(\hat{\jmath}\)) is treated separately, applying integral calculus to each individually.

This use of integrals allows us to link force-time data directly with momentum changes over time, providing deeper insight into evolving systems.

Vector Components

When dealing with physics problems involving vectors, breaking them into their components is key. Vectors like force and momentum have magnitude and direction, often represented in three axes: \(\hat{\imath}\), \(\hat{\jmath}\), and \(\hat{k}\).
In our exercise, all motion is horizontal and vertical, so we only consider the \(\hat{\imath}\) and \(\hat{\jmath}\) components.
To solve for momentum, each component is calculated separately:

• Vector components allow for simpler mathematical manipulation and integration.
• This approach helps tackle forces or motions acting in different directions independently.

By treating the \(\hat{\imath}\) and \(\hat{\jmath}\) directions separately, we apply force laws to each axis, then combine results to form the total momentum vector.

Net Force Analysis

Understanding net force is crucial for analyzing motion. Net force is the vector sum of all forces acting on an object. It dictates how an object moves according to Newton's Second Law of Motion.
In our problem, the net force changes over time, which influences how quickly or slowly the box's momentum changes.
Key points about net force include:

• It determines the acceleration and subsequently the change in velocity and momentum of a body.
• Time-varying forces, like in our exercise, require calculus to precisely describe their effect over a specific duration.
• By integrating these forces, we derive changes in momentum, essential for dynamics predictions.

Performing net force analysis provides insight into real-world applications, predicting how multiple forces combine to affect an object's motion.