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)