Problem 66 - University Physics with Modern Physics

Question

CP CALC A young girl with mass 40.0 \(\mathrm{kg}\) is sliding on a horizontal, frictionless surface with an initial momentum that is due east and that has magnitude 90.0 \(\mathrm{kg} \cdot \mathrm{m} / \mathrm{s} .\) Starting at \(t=0, \mathrm{a}\) net force with magnitude \(F=(8.20 \mathrm{N} / \mathrm{s}) t\) and direction due west is applied to the girl. (a) At what value of \(t\) does the girl have a westward momentum of magnitude 60.0 \(\mathrm{kg} \cdot \mathrm{m} / \mathrm{s} ?\) (b) How much work has been done on the girl by the force in the time interval from \(t=0\) to the time calculated in part (a)? (c) What is the magnitude of the acceleration of the girl at the time calculated in part (a)?

Step-by-Step Solution

Answer

(a) t \( \approx 6.05 \) s. (b) Work \( = 101.25 \) J. (c) Acceleration \( = -1.24 \) m/s².

Step 1: Initial and Final Momentum Analysis

The initial momentum of the girl is given as 90.0 kg·m/s due east. We want the westward momentum to be 60.0 kg·m/s. Since east and west are opposite directions, we will set the final momentum as -60.0 kg·m/s (negative sign indicates opposite direction to initial). Hence, applying conservation of momentum:\[ p_{final} = p_{initial} + (F \cdot t) \]Where \(F(t) = -8.20 \cdot t\) N (negative since the force is westward).

Step 2: Solving for Time (t)

Set the equation for momentum change:\[ -60.0 = 90.0 - (8.20 \cdot t^2 / 2) \]Because the force given is a function of time, the impulse (change in momentum) due to this force is the integral of force over time, which gives 8.20 \(t^2/2\) if F(t) is time-variable:\[ 150.0 = 4.10 \cdot t^2 \]Solving for \(t\):\[ t^2 = \frac{150.0}{4.10} \approx 36.59 \]\[ t \approx 6.05 \text{ seconds} \]

Step 3: Calculating Work Done by Net Force

The work done by the force is given by the integral of the force applied over the displacement. For this equation, Work \(W\) is:\[ W = \int_{0}^{6.05} F(t) \cdot v(t) \cdot dt \]Assuming average velocity considering initial and zero final speed, velocity at any point can be given from momentum/ mass due to Newton's second law:The net work done is also equal to the change in kinetic energy:\[ W = \Delta KE = \frac{p^2_{initial}}{2m} - \frac{p_{final}^2}{2m} \]Plug in the numbers:\[ W = \frac{90^2}{80} - \frac{60^2}{80} = 101.25 \text{ J} \]

Step 4: Calculating Acceleration at Time t

At time \(t = 6.05\) s, the acceleration \(a(t)\) can be calculated using Newton's second law, \(F = ma\).Forces:\[ F(t) = -8.20 \times 6.05 = -49.61 \text{ N} \] (as it is variable with time, at this specific instance)Mass = 40.0 kg, thus solving for acceleration:\[ a(t) = \frac{F(t)}{m} = \frac{-49.61}{40} = -1.24 \text{ m/s}^2 \]

Key Concepts

Net Force

Understanding net force is crucial when dealing with problems involving motion and change of momentum. Net force is essentially the total force acting on an object when you consider all the forces applied in different directions. It determines whether an object will accelerate, slow down, or stay in uniform motion.
In the given exercise, the girl is experiencing a net force that changes with time, given by the equation \( F(t) = (8.20 \, \text{N/s}) \times t \). Since this force is directed westward, it is applied opposite to her initial eastward momentum. This opposition results in a change in the direction of motion.
This concept of net force is key when calculating how a force changes an object's movement over time. The variable force in this exercise is applied over 6.05 seconds, changing the direction of the girl's momentum from east to west.

Work-Energy Principle

The work-energy principle tells us that the work done on an object by an external force changes its kinetic energy. In simpler terms, work done is the energy transferred to or from an object via the application of force along a displacement.
In the exercise, we calculated the work done on the girl by using the change in kinetic energy formula:

• The initial kinetic energy is determined by her eastward momentum of 90.0 kg·m/s.
• The westward final momentum of 60.0 kg·m/s gets factored in as well, representing a change in kinetic energy by the time the force stops.

The work done, which equates to 101.25 Joules, is the net energy change due to the applied net force. This calculation illustrates how a time-variable force can do work over time, changing the object's kinetic state.

Acceleration Calculation

Acceleration is the rate of change of velocity of an object, defined through Newton's second law. When there is a net force acting on a mass, the object will accelerate. In mathematical terms, this relationship is given by \( a = \frac{F}{m} \).
For the girl in the problem, the calculation of acceleration at the critical time of 6.05 seconds involves pinpointing the net force at that moment, \( F(t) = -49.61 \, \text{N} \). Here, the negative sign indicates the westward direction. With her mass given as 40.0 kg, we can find her acceleration by dividing the net force by her mass:

• Acceleration, \( a(t) = \frac{-49.61}{40} = -1.24 \, \text{m/s}^2 \)

This calculated acceleration shows exactly how quickly her velocity changes as the force is applied, reflecting the net result of the opposing force overcoming her initial eastward momentum.