Hi i need help with dynamics hw

MEGR 3121, Dynamics Systems – I

HW # 3, Due: March 10, 2014

Max: 360

Q1) A particle when gently nudged from rest at point A begins to slide on the surface of a

transparent umbrella as shown. Its motion is such that its speed increases uniformly at 0.5

m/s per second.

i) Find the time it took for the particle to reach B. ii) Determine its velocity and acceleration at point B in en, et system iii) Express his velocity and acceleration at B in i, j system (Hint: Recall dy/dx is

the slope of a tangent at a point on the curve; | ⃗ | the speed ) iv) Express its velocity and acceleration at B in er, eθ system (O is the origin of

the er, eθ system)

v) At the instant it is at point B, find the values of ̇ ̇ ̈ ̈ with respect to the polar coordinate system fixed at O (60)

Q2) A car travels along the level curved road with a speed which is decreasing at the constant rate of 0.6 m/s each second. The speed of the car as it passes point A is 16 m/s.

Calculate the magnitude of the total acceleration of the car as it passes point B which is 120

m along the road from A. The radius of curvature of the road at B is 60 m. (25)

Q3) A particle is constrained to move in a parabolic slot cut in a rectangular plate [12 m x

16 m] as shown. The equation of the slot is given by y 2 = x-2 [x,y in meters measured

from O]. Further, as the particle moves in the slot, its y-coordinate is constrained to

change uniformly at the rate of 0.25 m/s. At the instant shown (i.e. when x = 6 m),

i) Compute the radius of curvature

ii) Express its velocity and acceleration in i, j system (Hint: ⃗ ̇ ̇ , ̇ is given and ̇ could be found as you know the equation of the path)

y = 2-x 2

0.75 m

β

A B

X

Y

O

r

iii) Express its velocity and acceleration in en, et system

iv) Find the values of ̇ ̇ ̈ ̈ recorded from O. (60)

Q4) A ball is thrown vertically upward from point A with an initial speed of 100 ft/sec

and its acceleration (in ft/s 2 ) along its path is given by , k = 0.002 ft

-1 and v

is speed in ft/sec.

i) Find the maximum height to which the ball will ascend. Denote this by H.

ii) Compute ̇ ̇ ̈ ̈ (recorded relative to the stationary observer at O) when the ball reaches a height H/2 (i.e. half its maximum height)

iii) Compute the normal and tangential components of its acceleration when it reaches a height of H/2. What is the radius of curvature of its path at this

instant? (45)

Q5) For the illustrated pulley-system, derive an equation which relates the acceleration of

block B to the acceleration of block A. (30)

100

ft/s

50 ft

O θ

r H

0.5H

Path

A

Y

X

12 m

6 m

O

r

θ

y

x

Q6) Danica traveling at a speed of 200 km/hr on a straight road applies brakes to her car

at point A and reduces her speed at a uniform rate to 150 km/hr at C in a total distance of

300 meters.

a) Calculate the magnitude of the total acceleration of her car immediately after she passes point B. Assume BC to be a part of a circular path. [So, AB is a straight

line while BC is a circular arc.]

b) Compute the time she took to go from A to C. (20)

Q7) A ball tied to a thread is swung around (ccw) in a circular path as shown. It starts

with a speed of 2 m/s at point A and increases its speed uniformly at the rate of 0.5 m/s 2

to reach B. l = 0.5 m

a) Compute the time it took for the ball to go from A to B b) Compute the acceleration (magnitude and direction) of the ball at B c) Compute the average velocity of the ball for its path from A to B d) Compute the average acceleration of the ball for its path from A to B (40)

Q8) During a portion of a vertical loop, an airplane flies in an arc of radius λ = 600 m

with a constant speed of 400 km/h. When the airplane is at A (as shown), its velocity

vector makes an angle β = 30° with the horizontal, and radar tracking data is r = 800 m

and θ = 30°. Compute the values of ̇ ̈ ̇ ̈ recorded by the radar at this instant. [unit vectors for radar coordinate system are shown] (40)

A

B l

30°

30°

Q9) The particle P moves along the curved slot, a portion of which is shown. Its distance

in meters along the slot is given by s = t 2 /4, where t is in seconds. The particle is at A

when t = 2 s and at B when t = 2.2 s. Determine the average acceleration of P between A

and B. (20)

Q10) (20)

er

eθ

v