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Chapter 4: Problem 111

'The mass of an object changes with time according to \(m(t)=m_{0} e^{-b t}\), where \(m_{0}\) is the initial mass and \(b\) is a proportionality constant with units of \(s^{-1}\). The velocity of the object also changes with time, according to \(v(t)=a t+v_{0}\), where \(v_{0}\) is the initial velocity and \(a\) is the object's constant acceleration. (a) Determine an expression for the force on the object at any time \(t\). (b) Determine the force when \(m_{0}=2 \mathrm{~kg}, b=0.16 \mathrm{~s}^{-1}\), \(v_{0}=1 \mathrm{~m} / \mathrm{s}, a=6 \mathrm{~m} / \mathrm{s}^{2}\), and \(t=3 \mathrm{~s}\).

Short Answer

Expert verified
The expression for the force on the object at any time \(t\) is \(F(t)= m_{0}ae^{-bt} - m_{0}abte^{-bt} - m_{0}bv_{0}e^{-bt}\). The force when \(m_{0}=2 kg\), \(b=0.16 s^{-1}\), \(v_{0}=1 m/s\), \(a=6 m/s^2\), and \(t=3s\) is approximately 8.05 N.

Step by step solution

01

Recognize the Force-Mass-Acceleration Relationship

Start from the definition of force, which is given as \(F=ma\). Here, \(m\) is mass and \(a\) is acceleration. In this case, both \(m\) and \(a\) are functions of time, \(t\). So, \(F(t)=m(t)a(t)\).
02

Differentiate the Mass Function

Since mass is changing with time, we need to consider the rate of change of mass. Differentiate the mass function with respect to time, yielding: \(m'(t)= -m_{0}b e^{-bt}\).
03

Substitute Mass and its Derivative into the Force Equation

Substitute \(m(t)\) and \(m'(t)\) into Newton's Second Law: \(F(t)= m(t)a + v(t)m'(t)\). To find acceleration, differentiate the velocity function with respect to time. \(a(t) = \frac{dv}{dt} = a\). Thus, the equation becomes \(F(t)= m_{0}e^{-bt}a + (at + v_{0})(-m_{0}be^{-bt})\). Simplifying this we have: \(F(t)= m_{0}ae^{-bt} - m_{0}abte^{-bt} - m_{0}bv_{0}e^{-bt}\).
04

Substitute Values to Determine the Force at Specific Time

Now, substitute the given values into the equation to find the force at \(t=3s\). So, \(F(3)=2 \times 6 \times e^{-0.16 \times 3} - 2 \times 0.16 \times 6 \times 3 \times e^{-0.16 \times 3} - 2 \times 0.16 \times 1 \times e^{-0.16 \times 3}\). Using a calculator, the answer is approximately 8.05 N.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Newton's Second Law
Newton's Second Law is a cornerstone in understanding motion and dynamics. Isaac Newton articulated this law to describe the relationship between an object's motion and the forces acting upon it. The law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. This relationship can be expressed mathematically as: \[ F = ma \] In scenarios where the mass of an object varies over time, as in the given problem, the concept remains the same but requires careful consideration of how both mass and acceleration change with time. This ensures the accurate analysis of the resultant force.
Differentiation in Physics
Differentiation is a mathematical tool used to determine the rate at which a quantity changes. In physics, differentiation is crucial when dealing with changing quantities, such as velocity, acceleration, and mass. - When the mass of an object changes with time, it is essential to compute the derivative of the mass function. This derivative represents how quickly the mass is changing, which can affect the force calculation.- In the given exercise, the mass function is expressed as: \[ m(t) = m_0 e^{-bt} \] The differentiation results in: \[ m'(t) = -m_0 b e^{-bt} \] Differentiation also helps find the object's acceleration, making it fundamental in employing Newton's Second Law effectively.
Force Calculation
Calculating force in a dynamic system where mass and velocity change with time involves substituting the differentiated values into the fundamental force equation. In this exercise, the force is calculated as a function of time, considering changing mass and velocity. To perform the force calculation, utilize the modified version of Newton's Second Law: \[ F(t) = m(t)a(t) + v(t)m'(t) \] Substitute the expressions for mass and its derivative into the equation to derive the force function for any given time. Inserting specific values provided in the problem, such as initial mass, velocity, and constants, allows us to calculate the force at a particular instant (e.g., at \( t = 3 \) seconds). This process highlights the importance of meticulous calculations to ensure accurate results.

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Most popular questions from this chapter

You apply a \(60-\mathrm{N}\) force to push a box across the floor at constant speed. If you increase the applied force to \(80 \mathrm{~N}\), will the box speed up to some new constant speed or will it continue to speed up indefinitely? Assume that the floor is horizontal and the surface is uniform. Explain your answer.

-Sports A boxer claims that Newton's third law helps him while boxing. He says that during a boxing match, the force that his jaw feels is the same as the force that his opponent's fist feels (when the opponent is doing the punching). Therefore, his opponent will feel the same force as he feels and he will be able to fight on without any problems, no matter how many punches he receives or gives. It will always be an "even fight." Discuss any flaws in his reasoning.

Tension is a very common force in day-to-day life. Identify five ordinary situations that involve the force of tension.

Use a spreadsheet or graphing calculator to plot the velocity versus height of an express elevator. The elevator starts from rest on the ground floor, accelerates up to its maximum speed of \(10 \mathrm{~m} / \mathrm{s}\) after reaching the 4th floor (the distance between each floor is \(3.5 \mathrm{~m}\) ), remains at this speed until the 20th floor, and then stops at the 30th floor. Identify the point(s) on the graph when the weight of a rider will reach its maximum value. SSM

An astronaut has a mass of \(80 \mathrm{~kg}\). How much would the astronaut weigh on Mars where surface gravity is \(38 \%\) of that on Earth?
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