Dynamics

Assignment Motivation

The goal of this assignment is for students to review fundamental mathematical concepts as well gain as a deeper understanding of rectilinear motion.

Notes

• Vector notation can change depending upon the convention chosen. In this book, vectors are typically written as bold letters. I cannot write in bold on the board or on paper, and therefore I will use the arrow notation for vectors, p ≡ ??? , and the hat notation for unit vectors, ???? ≡ u^??.

Assignment

1. A vector can also be expressed explicitly using unit vectors as p^→= aiˆ + bˆj + ckˆ or as a column vector,

where the unit vectors are implicit. Given the explicit or implicit forms, determine the other:

a.

b. r^→= -8j^+k^

2. A vector also has magnitude,||p^→||, and direction, u_p^ = p^→/||p||, where the caret represents a unit vector. From this definition, a vector can be written as p^→= u_p^||p||. Express each vector as the product of a scalar (magnitude) times a unit vector (direction):
a. w^→= 2i^ - 1j^ + 3k^

b.

3. Vectors are typically visualized as an arrow, but calculations are performed using their components. Decomposing vectors into components is usually done in rectangular components (but as we will see, NOT always!). For the vectors given in the picture below:

a. Decompose a^→ into its x and y components if ||a^→||= 20m/s and express it in explicit form.

b. Decompose b^→ into its x and y components if ||b^→||= 15m/s and express it in implicit form.

4. Addition of vectors can be easily done when the vectors are expressed in component form by adding like directions together. For the following vectors a^→= -j^+k^, b^→=i^+4j^, c^→= 2i^+3j^-k^ Perform the following operations:
a. a^→+b^→
b. b^→+^c→
c. a^→-c^→

5. There are numerous notations for the derivative. The most common is the "d/dt notation." There are also other notations that are utilized in dynamics. The most common is the "dot notation" where the differentiation is assumed to be with respect to time:

y^.≡dy/dt, f^..≡d²f/dt²,...

Compute the following:

a. f^. where f(t) = 3t² + 2t - sin(t)
b. y^.. where y(t) = t cos(t) + 2t

6. The chain rule can be applied in dynamics with functions that are implicit dependent upon time. For example, a position of an object in polar coordinates could be a function of its angle which is also a function of time. That is: r(t) = f (θ(t))

Differentiation of this function requires the chain rule. Thus:

r^.= (df/dθ)θ^., r^..= (d²f/dθ²)θ^.+ (df/dθ)θ^..,

Compute the following:

a. r^. where r = 10sinθ and r = r(t), θ=θ(t)

b. r^.. where r = 5cosθ and r = r(t), θ=θ(t)

7. If s = (3t² + 2) m, determine v when t = 2s.

8. A particle moves along a straight line such that its position is defined by s = (t³ -3t² +2) meters. Determine the average velocity and the acceleration of the particle when t = 4 seconds.

9. A particle is moving along a straight line such that its acceleration is defined as ?? = (-2v) m/s^², where v is in m/s. If v= 20 m/s when s = 0 and t = 0, determine the particle's velocity as a function of position and distance the particle moves before it stops.

10. The acceleration of a rocket traveling upward is given by ?? = (6+0.02s) m/s², where s is in meters. Determine the time needed for the rocket to reach an altitude of s = 100 meters. Initially, v = 0 and s = 0 when t = 0.