Inertia
Definition of inertia (not the equation)
What is resistance to a change in motion?
Equation for the inertia of a single-point mass
What is I = mr2 (m = mass, r = radius/distance from axis of rotation).
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).
Using a long, thin, handle screwdriver is better for prying open a can of paint, why?
What is torque? The long handle = longer lever arm in relation to prying off the lid, and more torque
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
The total 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.
The inertia of a merry-go-round that has a mass of 2500-kg and a radius of 2.5 meters. (I = 1/2mr2 for a solid cylinder).
What is 7208 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?
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)
Why a gymnast might stretch their arms out to help them balance on a beam
What is increased radius and increased inertia?
For a single-point mass, the angular momentum is equal to the linear momentum times this.
What is the radius? L = mvr
A block of ice and a sphere of same mass are at the top of the same incline. The block of ice slides and the sphere rolls without slipping. What is the change in angular momentum of the sphere as it rolls without slipping from the top of the incline?
No change in angular momentum
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)?
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)
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
I * delta w = torque * delta time = mass * velocity * radius.
What is "impulse"?
A block of ice and a sphere of same mass are at the top of the same incline. The block of ice slides and the sphere rolls without slipping. Which of the two reaches the ground first?
Block of ice
The order of the following objects (assuming their mass and radii are similar): a medicine ball, a dumbbell weight, and a large tire (like you would do tire flips with).
What is increasing rotational inertia? A solid sphere has the LEAST rotational inertia because its 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.
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
A string wrapped around a pulley with a radius of 0.2 m and an inertia of .25 kg*m2. The string is pulled with a force of 15 N, resulting in this angular acceleration.
What is 12 rad/s2? Torque = Force x radius Torque = Inertia x angular acceleration so Force x radius = Inertia x angular acceleration
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.