Calculus 3 vector analysis question (Newton's 2nd problem)

In summary, the problem discussed is about calculating the force on an object traveling along the parabola y = x^{2} at a constant speed of 10 units/sec, due to its acceleration at the point (0,0). The solution presented in class involves finding the tangential and normal unit vectors, determining the curvature at the given point, and then using the mass and curvature to calculate the force. However, there is confusion regarding the derivation of the tangential unit vector, as it conflicts with the given fact that the speed is constant. The correct parameterization should be r = x i + x^{2} j, which leads to a different solution.
  • #1
chongj12
3
0
This problem was discussed in my calculus 3 class, and there is one step that I don't understand.

Homework Statement


An object of mass m travels along the parabola y = x[tex]^{2}[/tex] with a constant speed of 10 units/sec. What is the force on the object due to its acceleration at (0,0) (write the answer in terms of unit vectors i,j,k)? (this problem is from Thomas' calculus 11th edition, section 13.5 #20)

Homework Equations


f(x) = x[tex]^{2}[/tex]
a = atT + anN
at = d(|v|)/dt
an = [tex]kappa[/tex]|v|[tex]^{2}[/tex]
T = v/|v|
[tex]kappa[/tex] = |f''(x)|/((1+f'(x)[tex]^{2}[/tex])[tex]^{3/2}[/tex])
r = ti + t[tex]^{2}[/tex]j
=> v = i+2tj
=> |v| = (1+4t[tex]^{2}[/tex])[tex]^{1/2}[/tex]
N = (d(T)/dt)/(|d(T)/dt|)3. The Attempt at a Solution **** this was the solution presented in class:
at = 0 because d(10)/dt = 0
an = 100[tex]kappa[/tex]
[tex]kappa[/tex] = 2 at the point (0,0)
*******T = 1/((1+4t[tex]^{2}[/tex])[tex]^{1/2}[/tex]) (i+2tj)
T = i at the point (0,0)
N = j at the point (0,0)

=> F = ma = m(200)jThe step that I don't understand is marked with stars (Deriving T). Since the speed is given as 10 units/sec, shouldn't that be used as |v| rather than (1+4t[tex]^{2}[/tex])[tex]^{1/2}[/tex]? And if |v| = (1+4t[tex]^{2}[/tex])[tex]^{1/2}[/tex], doesn't this contradict the problem statement, which says that speed is constant? This makes little sense to me because we used different values for |v| throughout the problem
 
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  • #2
The source of your problem is here:
chongj12 said:
r = ti + t[tex]^{2}[/tex]j
=> v = i+2tj
=> |v| = (1+4t[tex]^{2}[/tex])[tex]^{1/2}[/tex]
This parameterization conflicts with the given fact that the speed is constant. Since you are given that y=x2, you do know that
[tex]\boldsymbol r = x \hat{\boldsymbol i} + x^2\hat{\boldsymbol j}[/tex]
See where this leads you.
 

Related to Calculus 3 vector analysis question (Newton's 2nd problem)

1. What is the purpose of using vector analysis in Calculus 3?

Vector analysis is a mathematical tool used to analyze and solve problems involving vectors, which are quantities that have both magnitude and direction. In Calculus 3, vector analysis is used to study and understand the behavior of objects in three-dimensional space, and to solve problems related to motion, forces, and other physical phenomena.

2. What is Newton's 2nd problem in Calculus 3 vector analysis?

Newton's 2nd problem, also known as the Second Law of Motion, is a fundamental principle in physics that states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In Calculus 3 vector analysis, this problem is often used to analyze the motion of objects in three-dimensional space, taking into account the forces acting on the object.

3. How do you approach solving a problem involving Newton's 2nd problem in Calculus 3 vector analysis?

To solve a problem involving Newton's 2nd problem in Calculus 3 vector analysis, you first need to identify the forces acting on the object and their respective magnitudes and directions. Then, you can use vector algebra and calculus techniques to find the acceleration of the object and its position and velocity at any given time. It is also important to carefully consider the coordinate system and units being used in the problem.

4. Can you provide an example of a problem involving Newton's 2nd problem in Calculus 3 vector analysis?

Sure, here's an example: A 2 kg object is suspended by a rope and is being pulled with a force of 20 N at an angle of 30 degrees above the horizontal. What is the acceleration of the object and its position after 3 seconds? To solve this problem, we would need to resolve the force vector into its x and y components, use Newton's 2nd law to find the acceleration, and then use equations of motion to find the position of the object at t=3 seconds.

5. How does understanding vector analysis and Newton's 2nd problem benefit other areas of science and engineering?

Vector analysis and Newton's 2nd problem are fundamental concepts in physics and are used in many areas of science and engineering, including mechanics, electromagnetism, and fluid dynamics. Understanding these concepts allows for a deeper understanding of the physical world and the ability to solve complex problems involving motion, forces, and energy. It also provides a basis for understanding more advanced concepts in mathematics and physics, such as differential equations and Lagrangian mechanics.

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