Inertia
Definition of inertia. (not the equation)
Resistance to a change in motion
Difficulty in forcing an acceleration
What is the equation for Newton's 2nd Law in terms of rotation.
tau=Ialpha
It was easier to balance the ruler when the mass was placed at the far end than when it was at the middle...why?
The difference in rotational inertia.
Mass at the end has a greater inertia, and is thus more resistant to changes in motion, so once it was balanced it was easier to keep balanced. Putting the mass in the middle decreases the inertia, which would make the ruler easier to rotate, but not easier to balance (balancing is the same as keeping the object at rest)
This law of linear momentum also applies to angular momentum:
What is the Conservation of Momentum?
The rotational kinetic energy of a figure skater is quadrupled when this is doubled.
What is the radius? (as represented by spreading out her arms: if radius doubles, K is quadrupled)
OR
What is angular speed?
K=1/2 Iomega^2
Equation for the moment of inertia (or rotational inertia) of a single-point mass.
I = summr^2
What does Newton's 1st Law mean in terms of rotational motion
"an object that is rotating will continue to rotate and an object at rest will stay at rest, unless a net torque acts on the object". (i.e. a change in rotational motion is cause by the application of a rotating force, torque).
Using a long, thin, handle screwdriver is better for prying open a can of paint, while using a short, thick handle screwdriver is better for unscrewing a rusted screw because of this:
What is torque? The long handle = longer lever arm in relation to prying off the lid, and more torque The thick handle = wider radius, which is the "lever arm" for the screw, thus more torque
This is the equation that represents conservation of angular momentum.
What is
I_1omega_1=I_2omega_2
The momentum before is equal to the momentum after
L=Iomega
The kinetic energy of a rolling can of soup when it reaches the bottom of an incline is (in general terms):
1/2mv^2+1/2Iomega^2
The inertia of a merry-go-round that has a mass of 2500-kg and a radius of 2.5 meters. (
I = 1/2mr^2
for a solid cylinder).
7,812 N*m =
I = 1/2(2500)(2.5)^2
If you want to make the tea-cup ride at an amusement park spin 4 times as fast (increasing angular acceleration), by what factor must you increase the torque applied to the spinning mechanism?
4 times. Newton's 2nd Law tells us that
tau=Ialpha
... so torque and angular acceleration are proportional, and the inertia of the ride is constant. So if you increase acceleration by a factor of 4, torque will also increase by a factor of 4.... 4(torque) = inertia x 4(angular acceleration)
Why does a gymnast stretch their arms out to help them balance on a beam?
Increased radius=increased inertia
For a single-point mass, the angular momentum is equal to the linear momentum times what?
The radius.
L=mvr
This is the equation you would use to determine the amount of power produced by the spinning fan of a boat motor.
Power is the rate at which work is done. For rotational motion,
W=tau/_\theta
(/_\theta)/t=omega
P=(tau/_\theta)/t=tauomega
How does a figure skater or diver reduce their inertia, which allows them to increase their angular velocity during flips and spins?
Reduce their radius by pulling their arms (and legs) closer to their body (center of mass)
A merry-go-round accelerates uniformly from rest and reaches an angular speed of 0.4 rad/s in the first 10 seconds. If the net applied torque is 2,000N, this is the moment of inertia for the merry-go-round.
What is 50,000 kg*m^2?
tau=Ialpha
acceleration is change in angular velocity over time (0.4 rad/s divided by 10 sec = 0.04 rad/s^2). Plug angular acceleration and torque into the equation to solve
Three children try and pull on a merry-go-round to make it rotate. The merry-go-round has a radius of 3.65 m. Two children pull with equal forces of 40N to the right (clockwise) and a third child tries to rotate the merry-go-round the other direction with a 60 N force at the same time. What is the net torque on the merry-go-round (including the direction it will turn)?
-73 N*m (clockwise motion is negative)
What does this equation represent?
I/_\w = tau/_\ t = mvr
"Impulse-momentum theory"
Why do racing bikes have large-thin tires and off-road bikes have smaller-thicker tires (assume the actual mass of a tire is about the same).
What is rotational inertia and Newton's 1st law? The greater the inertia, the greater the resistance to change and the easier to balance.... so a large thin tire distributes the mass further away from the radius, while the off-road tire puts the mass closer to the center... racing bikes are meant to go fast on smooth, flat surfaces while off-road bikes need to be able to adjust to changing terrain, and a lower inertia means that it will be easier for you to change the motion of the tire.
Order of the following objects (assuming their mass and radii are identical) by increasing rotational inertia: a medicine ball (solid sphere), a basketball (hollow sphere), and a large tire (hoop - perp to aor)
A solid sphere, like a medicine ball, has the LEAST rotational inertia because its mass is the most evenly distributed. The tire has the MOST inertia because all of the mass is concentrated at the maximum distance from the center.
A string is wrapped around a pulley with a radius of 0.05 m and an inertia of 0.2 kg*m2. If the string is pulled with a force, F, the resulting angular acceleration of the pulley is 2 rad/s2. What is the force applied?
8N
tau=Ialpha=Frsintheta
Set
Ialpha=Fr
and solve for force
A string wrapped around a pulley with a radius of 0.2 m and an inertia of .25 kg*m2. If the string is pulled with a force of 15 N, what is the angular acceleration?
12 rad/s2
tau=Fr=Ialpha
How is a planet's angular momentum conserved even though it travels around the sun in an elliptical orbit (meaning that r changes)?
Decreasing angular velocity the closer the planet gets to the sun.
I_1omega_1=I_2omega_2
... so the inertia of the planet increases as it gets closer to the sun, and the larger the inertia is, the smaller (or slower) the angular velocity must be to keep momentum conserved. A small inertia allows for greater angular velocity.
What are the names of my children?
Alyce & Dean