Circular Motion, Pt 2: Angular Dynamics

Mr. Velazquez

AP/Honors Physics

Formulas: Angular Kinematics

(𝜃 must be in radians): 𝑠 = 𝑟𝜃

𝜔 =𝜃

𝑡=

𝑣𝑡

𝑟

𝛼𝑎𝑣 =∆𝜔

𝑡=

𝑎𝑡

𝑡

𝑇 =2𝜋

𝜔=

1

𝑓

Arc Length

Angular Velocity

Angular Acceleration

Period of Rotation

360° = 2𝜋 rads = 1 rev

Formulas: Angular Kinematics

Tangential Quantities: 𝑣𝑡 = 𝑟𝜔 𝑎𝑡 = 𝑟𝛼

Linear Equation (𝒂 = constant)

Angular Equation (𝜶 = constant)

𝑣 = 𝑣0 + 𝑎𝑡 𝝎 = 𝝎𝟎 + 𝜶𝒕

𝑥 = 𝑥0 + 𝑣0𝑡 +1

2𝑎𝑡2 𝜽 = 𝜽𝟎 + 𝝎𝟎𝒕 +

𝟏

𝟐𝜶𝒕𝟐

𝑣2 = 𝑣02 + 2𝑎(𝑥 − 𝑥0) 𝝎𝟐 = 𝝎𝟎

𝟐 + 𝟐𝜶(𝜽 − 𝜽𝟎)

Centripetal Acceleration: 𝑎𝑐 =𝑣𝑡

2

𝑟= 𝑟𝜔2

Rolling Motion (no slipping)

• A perfectly round object that is rolling over a surface is experiencing static friction • The part of the object in contact with the surface is

always at rest

• The linear speed 𝒗 of the axle (center of the wheel) is therefore completely dependent on the angular velocity 𝝎 and the radius 𝒓 of the wheel.

𝑣 = 𝑟𝜔

Rolling Velocity (w/o slipping)

Example: Rolling Motion A bicycle slows down uniformly from a speed of 𝑣0 = 8.40 m/s to rest, over a distance of 115 m. Each wheel and tire has an overall diameter of 68.0 cm. Find: a) The initial angular velocity of the wheels b) The total number of revolutions made by each

wheel before coming to rest c) The angular acceleration of the wheels d) The time it took to come to a stop

115 m

𝑣0 = 8.40 m/s 𝑣 = 0

68

.0 c

m

Angular Dynamics: Torque & Moment • Causing rotation obviously requires a force (Newton’s 1st

Law)

• The direction and location of this force are highly important

• In the example below, a lever extends a certain length from the fulcrum (center of rotation). Several forces act on the lever.

• Assuming all three forces have the same magnitude, which force will cause the fastest rotation? Which force will not cause any rotation?

𝐹𝐴 𝐹𝐵

𝐹𝐶 fulcrum

fastest rotation

no rotation

Angular Dynamics: Torque & Moment

• 𝐹𝐵 will cause a greater angular acceleration for two very important reasons: • Its direction is perpendicular to the lever • It’s applied farther from the axis of rotation (fulcrum)

• 𝐹𝐶 will not cause any rotation, even though it is applied farther from the fulcrum than 𝐹𝐵 • This is because its direction is parallel to the lever • In order to cause rotation, some component of the force

must be perpendicular to the lever

𝐹𝐴 𝐹𝐵

𝐹𝐶 fulcrum

fastest rotation

no rotation

Angular Dynamics: Torque & Moment • We can now define torque 𝝉 as the product of the perpendicular

force acting on a lever and the distance away from the fulcrum this force is applied (called the lever arm) • Unit: 𝐦 ⋅ 𝐍 (meters times Newtons—not the same thing as a joule,

which is a N ⋅ m)

• The greater the torque caused by a force, the greater the resulting angular acceleration (direct proportion)

• If the force is applied at an angle 𝜽, we simply find the perpendicular component of that force and use it to calculate torque

𝑟 (“lever arm”)

𝐹

𝐹 sin 𝜃

𝐹 cos 𝜃

𝜃

𝜏 = 𝑟𝐹⊥

𝜏 = 𝑟𝐹 sin 𝜃

Torque

Example: Torque & Moment

Two circular disks of radii 𝑟𝐴 = 30 cm and 𝑟𝐵 = 50 cm are attached to each other on an axle that passes through the center of each. Calculate the net torque on this compound disk due to the two forces shown.

𝑟𝐵

𝑟𝐴

𝐹𝐴 = 50 N

𝐹𝐵 = 50 N

Angular Dynamics: Rotational Inertia

To see how torque relates to acceleration, let’s examine a simple example of a particle of mass 𝑚 rotating in a circle of radius 𝑟. A force 𝐹 is applied to the mass in a direction tangent to the circle.

𝑟 𝑚

𝐹 The applied force will cause tangential acceleration. So we can combine the equation for tangential acceleration with the equation for Newton’s Second Law:

𝐹 = 𝑚𝑎 𝐹 = 𝑚𝑟𝛼

Now, multiplying both sides by 𝑟:

𝑟𝐹 = 𝑚𝑟2𝛼 Torque (𝜏) Moment of Inertia (I)

∑𝜏 = 𝐼𝛼

Torque & Moment of Inertia

Angular Dynamics: Rotational Inertia

• The moment of inertia (I) is simple for a single particle in circular motion. But a rotating rigid object (like a wheel) consists of many particles, all at different distances from the center of rotation.

• Therefore, the moment of inertia for a rigid object depends on the shape and density of the object and the location of the axis of rotation.

• The total torque on an object is related to the total moment of inertia in the following way:

∑𝜏 = (∑𝑚𝑟2)𝛼

∑𝜏 = 𝐼𝛼

Angular Dynamics: Rotational Inertia

These values were computed using integral calculus; a similar table can be found in your book (Pg. 291, Table 10-1)

Example: Rotational Inertia Three forces are applied, as shown, to a solid circular disk of mass 3.50 kg and a radius of 25.0 cm (with an axis of rotation through the center) that is at rest. Calculate:

a) The net torque acting on the disk

b) The moment of inertia of the disk

c) The resulting angular acceleration 𝛼

25.0 cm

𝐹𝐴 = 60.0 N

𝐹𝐵 = 40.0 N

𝐹𝐵 = 55.0 N 30°

45°

Rotational Kinetic Energy

Any rigid, rotating object will have rotational kinetic energy (units are still joules):

𝐾𝑅 =1

2𝐼𝜔2

If the object also undergoes translational motion as it spins (think: a tire rolling down a ramp or ball rolling across a floor), then it will have both rotational and translational kinetic energies:

𝐾𝑡𝑜𝑡𝑎𝑙 = 𝐾 + 𝐾𝑅 =1

2𝑚𝑣2 +

1

2𝐼𝜔2

Rotational Kinetic Energy

Total Kinetic Energy

Rotational Kinetic Energy Just like the work done by a force can be thought of as the change in kinetic energy, we can also define the work done by torque as the change in rotational kinetic energy:

𝑊 = 𝜏∆𝜃 =1

2𝐼∆𝜔2

Work Done by Torque

This way, the work done by torque can be thought of as the work required to get an object spinning at a rate of 𝜔, starting from rest.

Example: Rotational Kinetic Energy

A spherical ball of mass 𝑀 and radius 𝑅 and a circular ring of the exact same mass and radius both sit at rest at the top of an incline at a vertical height of 𝐻. When they are released, which will reach the bottom of the incline first?

Angular Momentum

• When an object rotates or spins with angular speed 𝜔, we say that it has angular momentum (L), and we calculate it exactly as expected: by replacing 𝑚 and 𝑣 with their angular analogues 𝐼 and 𝜔:

𝐿 = 𝐼𝜔 Angular Momentum

• Just as with other angular vector quantities, angular momentum can be translated into a tangential quantity that depends on the distance 𝑟 from the center:

𝐿 = 𝑟𝑝 Angular and Linear Momentum

Conservation of Angular Momentum

• Angular momentum (just like linear momentum) will always be conserved.

• If net external torque is zero, it follows that final angular momentum will equal the initial angular momentum:

World Record Figure Skating Spin: https://www.youtube.com/watch?v=AQLtcEAG9v0

𝐿1 = 𝐿2 𝐼1𝜔1 = 𝐼2𝜔2

Conservation of Angular Momentum

Exit Ticket: Rotational Dynamics A small mass (𝑚 = 1.5 kg) attached to the end of a string revolves in a circle on a frictionless tabletop. The other end of the string passes through a hole in the table. Initially, the mass revolves with a tangential speed of 𝑣1 = 2.4 m/s in a circle of radius 𝑟1 = 0.80 m. The string is then pulled so that the radius is reduced to 𝑟2 = 0.48 m. Find:

a) The final speed 𝑣2 of the mass

b) The initial and final angular speeds 𝜔1 and 𝜔2

c) The total kinetic energy before and after the radius changes

Homework Problem Set: Rotational Dynamics

Pg. 302–303 #44–60 (mult of 4) Pg. 342–345 #4–32 AND #40-60

(mult of 4)