Transcript Example2 - mrdsample

Angle measurement can be
defined in degrees but also
can be defined in radians.
The angle θ in radians is
defined:
Angular Quantities
Angular displacement:
The average angular velocity is
defined as the total angular
displacement divided by time:
The angular acceleration is the rate at which the
angular velocity changes with time:
Converting between angular and linear
Every point on a rotating body has an angular
velocity ω and a linear velocity v.
If the angular velocity of a
rotating object changes, it
has a tangential
acceleration:
Even if the angular
velocity is constant, each
point on the object has a
centripetal acceleration:
Constant Angular Acceleration
The equations of motion for constant angular
acceleration are the same as those for linear
motion, with the substitution of the angular
quantities for the linear ones.
Example1
A record player turntable is turned on. It is noted that
the disk turns 0.75 revolutions as it goes from rest to
33.3 rpm, clockwise.
(a) Determine the disk’s final angular velocity in
radians per second.
(b) Determine the disk’s angular acceleration.
Example2
A cyclist is traveling at 7.0 m/s to the right when she
applies the brakes and slows to a speed of 5.0
m/s. Each wheel of the bicycle has radius 0.33 m and
completes 5.0 revolutions during this braking period.
Determine the time that elapses in this period.
Example3
A string is wrapped around the axle of a gyroscope –
30.0 cm of the string is in contact with the axle, which
has diameter 2.20 mm. Starting at rest the string is
pulled with a constant acceleration, which causes the
gyroscope to start spinning. It takes 1.10 seconds to
pull the string off of the axle and the gyroscope then
spins for an additional 60.0 seconds before stopping.
(a) Find the maximum angular speed of the gyroscope
in rad/s.
(b) Find the total number of revolutions the gyroscope
will spin.
Torque
To cause a change in rotation, a net
torque is required.
Torque is the force applied at a perpendicular
distance from the pivot of rotation.
Torque is defined as:
Calculate the net torque and direction about a
pivot at the center of the 1.0m long uniform beam
25N
30o
45o
20o
10N
30N
Rotational Inertia
The resistance to
change an object’s state
of rotation is called
rotational inertia.
Various shapes and
formulas for rotational
inertia. Depends on
mass distribution and
location of axis of
rotation.
General form of
equation is:
Rotational Dynamics
Just like a net force can cause an
acceleration according to FNET = ma
A net torque can cause a
rotational acceleration where
Example
A 150kg merry-go-round (treat as a solid disk)
of radius 1.50m is set in motion by pulling on
a rope that is wrapped around the rim. What
constant force must be exerted on the rope to
give the merry-go-round a speed of
0.50revs/sec in 2.0s?
Example2
A 5.0kg pulley with radius 0.60m and
frictionless axle starts from rest and
speeds up uniformly as a 3.0kg mass
falls making a light rope unwind from
the pulley. The mass falls for 4.0s.
Treat pulley as disk.
a) What is the linear acceleration of the mass?
b) How far does mass fall in the time stated?
+
c) What is the angular acceleration of the
pulley?
d) What is the tension in the rope?
Example3
A potters wheel has a radius of 0.50m and a
moment of inertia of 12kgm2 is rotating freely at
50rev/min. The potter can stop the wheel in 6.0s
by pressing a finger against the edge and
exerting a radial inward force of 70N. Find the
coefficient of friction between wheel and finger.
Rotational Kinetic Energy
Recall that the translational kinetic energy of a
moving object is given by
Total Kinetic Energy
Example
Find the angular speed of a solid sphere (R = 0.2m)
at the bottom of the 4.0m long incline if it starts from
rest at the top.
20o
Work & Rotational Kinetic Energy
The torque does work as it moves the wheel
through an angle θ:
Angular Momentum and Its Conservation
Recall that linear momentum was p = mv
A net torque causes a change in angular
momentum.
Systems that can change their rotational inertia through
internal forces will also change their rate of rotation
since ang momentum is conserved:
If the net torque on an
object is zero, the total
angular momentum is
constant.
A student (60kg) sits at the edge of spinning
merry-go-round (mass = 100kg, radius = 2.0m)
that spins on frictionless axle with speed 2.0rads/s.
Student then walks very slowly from edge towards
center. Find new angular speed when she reaches
0.50m from center. Must find Isys
Conditions for equilibrium
1)
2)
3)
Example1
A bridge weighs 2.23x106 N. On it are a tractor-trailer truck
that has mass of 14900kg and is positioned 9.75m from the
left end of the bridge. A car with mass 1590kg is located
10.80m from the right side of the bridge. A pickup truck
(2409kg) is positioned 5.25m from the right side of the bridge.
a) What is the upward force exerted by the pier on the right to
support the bridge.
b) What is the upward force exerted by the pier on the left to
support the bridge.
Example2
A 75-kg block is suspended from the end of a uniform
100-N beam. If θ = 30º, what are the values of T2 as
well as the horizontal and vertical forces on the hinge?
Example3
A 700N bear walks on a beam to get a basket of goodies. The
uniform beam weighs 200N and is 6.0m long. The goodies
weigh 80N. Find the tension in the wire and the components
of the reaction force at the hinge when the bear is at x = 1.0m
Example4
A ladder of length 8.0m and weight 350N is leaning
against a smooth wall at an angle of 60o with the
horizontal. A person of mass 90kg stands 3/4 of the
way up the ladder.
a) What frictional force does the ground need to apply to
prevent the ladder from sliding?
b) What is the minimum coefficient of static friction?
483.5N, 0.39
Example5
A 2.73-kg lamp is sitting on a table as shown. The lamp has a
base with a diameter of 18.0 cm, a height of 68.7 cm. The
coefficient of static friction between the base of the lamp and
the table is 0.32. A horizontal force, F, is applied to the central
column of the lamp at a height of h. What is the maximum
height above the base that the force, F, can act without toppling
the lamp? Note that pivot has been chosen.
Force diagram next
slide
FN