EP324 Assignment #1 1. The cone rolls without slipping such that at the instant shown

rad/s and rad/s2. Determine the velocity and acceleration of point A at this instant.

Sol) Problem of one moving cone (Origin at O) Necessary equations

(Eq. 1) (Eq. 2) From the conditions,

, (Fixed origin),

and

What to be determined: and

i) Angular velocity of the cone, (Similar with Ex.1)

(Rotation about z axis) (Spinning about a cone’s axis) At point P, (under no slipping condition)

or rad/s (along the spool axis)

Then, rad/s

y

z

P 2 ft

r2

A

40o

ii) Angular acceleration of the cone,

where ,

By inserting values of and into the equation for ,

Then,

ft/s (Answer) And

(Answer)

Because changes its direction due to

2. If the rod AB is attached with ball-and-socket joints to smooth collar A and B at its end points, determine the speed and acceleration of B and the angular velocity of the rod ( ) at the instant shown if A is moving downward with a speed of ft/s and an acceleration

ft/s2. The angular velocity of the rod is directed perpendicular to the rod’s axis.

Sol) Problem of one moving rod

Necessary equations

(1) Velocity

From the conditions, ft/s and ft (Position of B about A)

What to be determined: and Since collar B moves along the horizontal bar, (having only x-component) By expressing

(3 scalar equations)

(x-axis)

(y-axis) 3 equations for 4 unknowns

(z-axis)

Forth equation from the condition, (Perpendicular to )

By solving 4 equations with 4 unknowns,

rad/s, rad/s, rad/s

ft/s (along x-axis) (Answer)

(2) Acceleration From the conditions, ft/s From the previous problem 3, rad/s What to be determined: and Since the collar B moves along the horizontal bar, (having only x-component) By expressing

i.e. (1)

(2) ⇒ 3 Eqs. with 4 unknowns, but Solvable

(3)

(4)

and

ft/s2 (Answer)

3. At a given instant, the antenna has an angular motion rad/s and 2 rad/s2 about the z axis. At this same instant

, the angular motion about the x axis is 1.5 rad/s and 4 rad/s2. Determine the velocity and acceleration of the

signal horn A at this instant. The distance from O to A is d = 3 ft. Sol) Two rotating component ( and )

Necessary equations

1. Determine Velocity of the horn A,

Information from the problem description

(Fixed origin) & (Fixed distance from O to A)

rad/s & ft

Then,

∴ (Answer)

2. Determine Acceleration of the horn A,

Information from the description

: Fixed origin & : (Fixed distance between O to A)

Then,

∴ ft/s2 (Answer)

Since

4. Rod AB is attached to a disk and a collar by ball- and-socket joints. If the disk is rotating with an angular acceleration rad/s2, and at the instant shown has an angular velocity rad/s, determine the velocity and acceleration of the collar at A at the instant shown

Sol) 1. Velocity of A, Information from the description

ft/s & ft Use Eq. ( : NOT same as the angular velocity of disk, 2 rad/s)

where &

Then,

or x component:

y component:

z component: (3 equations, but 4 unkowns!) Since : Perpendicular to

(4th equation)

Solving these 4 equations yields:

rad/s &

ft/s (Answer)

α = 4 rad/s2

2. Acceleration of A, Use Eq.

where &

( : NOT same as the angular acceleration of disk, 4 rad/s)

and the information from the discription:

ft/s2 Let and Then,

or

or x component: (1)

y component: (2)

z component: (3)

(3 equations, but 4 unkowns!)

Solving these equations for aA yields: Eq.(1) – [Eq.(2)+Eq.(3)]/3

∴ ft/s2 (Answer)

5. At a given instant, the rod has the angular motions shown, while the collar C is moving down relative to the rod with a velocity of 6 ft/s and an acceleration of 2 ft/s2. Determine the collar’s velocity and acceleration at this instant.

Sol) 1. Velocity of collar C,

Necessary Eq. Information from the description (Collar C: located on y-z plane)

rad/s and (Fixed origin)

rad/s2

ft

Then, (Answer) 2. Acceleration of collar C,

Necessary Eq. Information from the description

rad/s2 and (Fixed origin)

Then,

∴ ft/s2 (Answer)