Show:
Inertia
Newton's 2nd Law
Rotational Dynamics
Two conditions for Equilibrium
Mystery Questions
100
Definition of inertia. (not the equation)
What is resistance to a change in motion (how stubborn an object is)?
100
The equation for Newton's 2nd Law in terms of rotation.
What is: torque = Inertia x angular acceleration
100
It was easier to balance the ruler when the mass was placed at the far end then when it was at the middle, why?
What is 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)
100

A uniform horizontal beam of mass M and length Lo is attached to a hinge at point P, with the opposite end supported by a cable, as shown in the figure. The angle between the beam and the cable is 0o. What is the magnitude of the torque that the cable exerts on the beam?

(mgLo)/2

100
The rotational kinetic energy of a figure skater is quadrupled when this is doubled.
What is a the radius (as represented by spreading out her arms? K = Iw, and I = mr^2; so 2r = 2^2 or 4 (if you double mass, kinetic energy is only doubled)
200
Equation for the inertia of a single-point mass.
What is I = mr^2 (m = mass, r = radius/distance from axis of rotation).
200
What Newton's 2nd Law means in terms of rotational motion.
What is "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).
200
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
200
This is the equation that represents conservation of angular momentum.
What is I(1)w(1) = I(2)w(2) - the momentum before is equal to the momentum after - and momentum is = inertia x angular velocity
200
The kinetic energy of a can of soup when it reaches the bottom of an incline.
What is 1/2 mv^2 + 1/2 Iw^2 ? The kinetic energy of an object that rotates down an incline is equal to the linear kinetic energy of the center of mass, and the rotational kinetic energy of the object.
300
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).
What is 5200 N*m? I = 1/3(2500)(2.5)^2
300
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?
What is quadrupled or 4-times? Newton's 2nd Law tells us that torque = inertia x angular acceleration... 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)
300
Why a gymnast might stretch their arms out to help them balance on a beam.
What is increased radius, and increased inertia?
300
For a single-point mass, the angular momentum is equal to the linear momentum times this.
What is the radius? L = mvr
300
This is the equation you would use to determine the amount of power produced by the spinning fan of a boat motor.
What is P = torque x angular displacement divided by time or torque x angular velocity Power is the rate at which work is done, and for rotational motion, work = torque x angular disparagement, and angular displacement over time is also angular velocity
400
How a figure skater or diver reduces their inertia, which allows them to increase their angular velocity during flips and spins.
What is reducing their radius by pulling their arms (and legs) closer to their body (center of mass)?
400
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 2000N, this is the moment of inertia for the merry-go-round.
What is 50,000 kg*m^2? Torque = inertia x angular acceleration, 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, inertia = torque/angular acceleration, 2000N/0.04 rad/s^2 = 50,000 kg*m^2)
400
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. This is the net torque on the merry-go-round (including the direction it will turn).
What 73 N*m, to the right (clockwise) - Net torque = sum of torques - counter clockwise is positive, and clockwise is negative. Net torque = -(3.65)(40) - (3.65)(40) + (3.65)(60) = - 73
400
I * delta w = torque * delta time = mass * velocity * radius.
What is "impulse-momentum theory"?
400
Reason why 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.
500
The order of the following objects (assuming their mass and radii are similar): a medicine ball, a basketball, a dumbbell weight, and a large tire (like you would do tire flips with).
What is increasing rotational inertia? A solid sphere, like a medicine ball, has the LEAST rotational inertia because it's mass is the most even distributed. The tire has the MOST inertia because all of the mass is concentrated at the maximum distance from the center.
500
A string is wrapped around a pulley with a radius of 0.05 m and an inertia of 0.2 kg*m^2. If the string is pulled with a force, F, the resulting angular acceleration of the pulley is 2 rad/s^2. This is the magnitude of the force.
What is 8 N? Torque = inertia x angular accleration torque = Force x radius x sin theta (and for a pulley, theta is 90, so sin90 = 1) set inertia x acceleration = force x radius and solve for force: force = (inertia x acceleration)/radius force = (0.2 x 2) / 0.05 = 8
500
A string wrapped around a pulley with a radius of 0.2 m and an inertia of .25 kg*m^2. The string is pulled with a force of 15 N, resulting in this angular acceleration.
What is 12 rad/s^2? Torque = Force x radius Torque = Inertia x angular acceleration so Force x radius = Inertia x angular acceleration
500
This is how a planet's angular momentum is conserved even though it travels around the sun in an elliptical orbit (meaning that r changes).
What is decreasing angular velocity the closer the planet gets to the sun? I1w1 = I2w2 ... so an 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.
500
Why the angular velocity of a planet changes as it moves closer/further from the sun.
What is conservation of angular momentum? Iw = Iw ... so as the planet gets closer, r gets smaller so I gets smaller and w must get faster/larger to conserve momentum.