Rotational Motion

M. Rocha   

Physics 1 - Week 4 - Chapter 8

Rotation vs. Revolution

Rotation: when an object turns about an internal axis—that is, an axis located within the body of the object

Revolution: when an object turns about an external axis.

Illustration demonstrating rotation around an internal axis
Illustration demonstrating revolution around an external axis

Rotation vs. Revolution

Diagram comparing rotation about an internal axis with revolution about an external axis

Angular Velocity

\mathrm{Angular \ Velocity} = \frac{\mathrm{Number \ of \ Rotations/Revolutions} }{ \mathrm{Time \ interval}}
\mathrm{Angular \ Velocity} = \omega = \frac{\mathrm{radians}}{t}

RPMs (Revolutions/Rotations per minute) is a commonly used measure of angular velocity

In Physics we like to use "radians per unit time" as a measure of angular velocity

Diagram defining one radian as the angle subtended by an arc equal in length to the circle radius
2 \pi \ \mathrm{radians} = \mathrm{1 \ full \ circle }
L = \theta \times r
Circle diagram showing arc length L equals angle theta in radians times radius r

Small angle approximation

Arc Length =

r \sin \theta \approx r \ \theta
\approx

r = Hypotenuse

Adjacent

Arc Length and The Small Angle Approximation

Diagram illustrating the small angle approximation comparing arc length to triangle side
Unit circle diagram showing standard angles in both degrees and radians

Circle Trigonometry 

\pi = 180^{\circ}
\pi/2 = 90 ^{\circ}
3\pi/2 = 270^{\circ}
2\pi = 360^{\circ}
\pi =\frac{\mathrm{Circunference}}{\mathrm{Diameter}} = \frac{\mathrm{Circunference}}{2 \times \mathrm{Radius}}

Checkpoint 1

\omega = \frac{\mathrm{radians}}{t}

If a race car travels around a circular track with an angular velocity of                                             , how long does it takes to complete a full revolution.

\omega = \pi \ \mathrm{radians/min}

2 minutes

2 \pi \ \mathrm{radians} = \mathrm{1 \ full \ circle }
Race car traveling around a circular track

r 

Tangential Velocity

Vt

Tangential velocity is the linear velocity of something moving along a circular path

v_\mathrm{tangential} = \omega \times r

If angular velocity is given in radians, then 

Diagram of a circular sector illustrating angle theta in radians
v_\mathrm{tangential} = \frac{\mathrm{arc \ length}}{\mathrm{time}}
= r \ \theta
\theta \mathrm{\ in \ radians}
Circular path diagram showing tangential velocity vector perpendicular to radius

Checkpoint 2

\omega = \frac{\mathrm{radians}}{t},

If car 1 and car 2 below travel on a circular track with the same angular velocity, which one has a larger tangential velocity and by how much? 

Car 1 travels 2x faster

v_t = \omega \times r

car 2

2r 

car 1

r

b)

Diagram of car 1 traveling on an outer circular track at radius 2r
Diagram of car 2 traveling on an inner circular track at radius r
Turntable diagram showing points at different radii rotating at the same angular speed but different linear tangential speeds

Tangential Velocity

All parts of the turntable rotate at the same angular velocity

A point farther away from the center travels a longer path in the same time and therefore has a greater tangential velocity

Rotational Inertia

Inertia: The tendency of an object to resist changes in velocity. An object at rest tends to stay at rest and an object in motion tends to remain moving in a straight line. 

Rotational Inertia: The tendency of a rotating object to resist changes in angular velocity. An object rotating about an axis tends to remain rotating about the same axis unless interfered with by some external force.

Rotational Inertia

Diagram showing mass distribution relative to the axis of rotation affecting rotational inertia

Rotational Inertia depends on the amount of mass and the distance of mass from the axis of rotation

Rotational Inertia

Tightrope walker holding a long balancing pole to increase rotational inertia
Diagram illustrating how increasing pole length increases rotational inertia and stability

The longest the pole the more balance it provides

Rotational Inertia depends on the amount of mass and the distance of mass from the axis of rotation

Rotational Inertia

Rotational Inertia depends on the amount of mass and the distance of mass from the axis of rotation

Cylinders of different mass distributions rolling down an incline to compare rotational inertia

Checkpoint 3

For an ice skater rotating, which body positioning has the greatest rotational inertia

d) has the most amount of mass distant to the axis of rotation, so it has the highest rotational intertia

Diagram showing four different body positions of a rotating figure skater (a, b, c, d) comparing mass distribution from the rotation axis

Consider balancing a hammer upright on the tip of your finger.  The end is heavier than the handle. Which way is it easier to balance? a) with the head at the top, or b) the other way around.

a) With the head at the top.

That way the rotational inertia is the greatest so it will be harder for it to rotate about your finger.

Diagram comparing balancing a hammer on a finger with head up versus head down

b)

a)

Torque

Illustration of applying force to a wrench showing lever arm distance multiplying rotational force

 You know that a longer lever arm makes it easier to rotate objects that require a lot of rotational force

Torque

Torque is rotational force. It is proportional to the applied tangential force and the distance to the axis of rotation.

\mathrm{Torque} = \mathrm{Force} \times \mathrm{lever \ arm \ length} = F \times r
Diagram defining torque as applied force times lever arm distance from the pivot

Angular Acceleration

A force causes a change in velocity, i.e. acceleration (Newton's 2nd law)

\mathrm{Angular \ Acceleration} = \frac{\mathrm{Torque}}{\mathrm{Rotational \ Inertia}} = \frac{ F \times r}{I}

A torque causes a change in angular velocity, i.e. angular acceleration (Newton's 2nd law in rotational motion)

a = \frac{F}{m}

Balanced Torques

\sum{\mathrm{Torques}} = 0 \Rightarrow \mathrm{No \ rotation}
Balanced seesaw diagram showing equal counterclockwise and clockwise torques resulting in zero net torque
(F \times r)_\mathrm{counter \ clockwise} - (F \times r)_\mathrm{clockwise} = 0

Checkpoint 4

If the system below is balanced (not rotating), what is the weight of the block hung at the 10-cm mark?

15 N

Meter stick balance problem showing unknown weight at 10-centimeter mark balancing known weights

Angular Momentum

Angular Momentum is rotational inertia in rotational motion

\mathrm{Momentum} = \mathrm{mass} \times \mathrm{velocity}
p = mv

Momentum is inertia in motion

\mathrm{Angular \ Momentum} = \mathrm{rotational \ inertia} \times \mathrm{rotational \ velocity}
\mathrm{Angular \ Momentum} = I \omega

Angular Momentum

Angular Momentum is rotational inertia in rotational motion

\mathrm{Angular \ Momentum} = \mathrm{rotational \ inertia} \times \mathrm{rotational \ velocity}
\mathrm{Angular \ Momentum} = I \omega

For a point mass

I \omega = m r^2 \omega = m r^2 \frac{v}{r} = m v r
Diagram of a point mass m rotating at distance r with tangential velocity v
I = m r^2 ,
\omega = \frac{v}{r}

Conservation of Angular Momentum

If no external net torque acts on a rotating system, the angular momentum of that system remains constant.

Demonstration of conservation of angular momentum: person on a rotating stool pulling hand weights inward to spin faster

Conservation of Angular Momentum

If no external net torque acts on a rotating system, the angular momentum of that system remains constant.

Cliff diver tucking and extending body to change rotational speed while in mid-air

Red Bull Cliff Diving 2015 

Conservation of Angular Momentum

If no external net torque acts on a rotating system, the angular momentum of that system remains constant.

Figure skater spinning slowly with arms extended and spinning rapidly when arms are brought in tightly

Figure Skating's Quintuple Jump Video

Centripetal Force

Object being whirled in a circular path by a string with tension providing centripetal force

Centripetal force is the force that keeps objects in circular motion

Washing machine drum spinning clothes to illustrate centripetal force in circular motion

Centripetal Force

Centripetal force always points to the center and its magnitude is

Diagram of an object in circular motion showing the inward centripetal force vector Fc directed toward the center
F_{c} = \frac{mv_{t}^2}{r}
F_{c}

Centripetal Force

Diagram of a car rounding a curve showing tire friction providing centripetal force and skidding when friction is insufficient

Centripetal force holds a car in a curved path.

a) For the car to go around a curve, there must be sufficient friction to provide the required centripetal force.

  b) If the force of friction is not great enough, skidding occurs.

Checkpoint 5

F_{c} = \frac{mv_{t}^2}{r}

A car with mass m = 1000 kg takes a turn with a radius of curvature of r = 30 m at a speed of v = 30 m/s (about 65 miles/h) , if the force of friction between the tires and the road is 7000 N, would it make the turn?

F_{c} = \frac{mv^2}{r} = \frac{1000kg \times (30 m/s)^2}{30m} = 30000 N

No, it will skid

Centripetal Force vs. Centrifugal Force

Illustration of a can whirled in a circle by a string showing inward centripetal force acting on the can

The only force that is exerted on the whirling can (neglecting gravity) is directed toward the center of circular motion. This is a centripetal force. No outward force acts on the can

Centripetal Force v.s. Centrifugal Force

Illustration of a ladybug inside a whirled can perceiving an outward centrifugal force from its rotating frame of reference

However from the reference frame of the lady bug it feels like there is a centrifugal force away from the center, and the centripetal force is just a support force

Centrifugal Force

Centripetal Force v.s. Centrifugal Force

The apparent centrifugal inside a rotating reference frame feels just like gravity, we can call this simulated gravity

Diagram comparing outside observer view of floor normal force with inside observer feeling fictitious centrifugal gravity in rotating frame

a) As seen from the outside, the only force exerted on the man is by the floor

b) As seen from the inside, there is a fictitious centrifugal force that simulates gravity

Simulated Gravity

Artwork of a cylindrical space colony habitat rotating to produce simulated gravity along its interior surface
Diagram of a rotating wheel space station generating artificial gravity through rotation
Cross-section diagram of a rotating space station showing outward centrifugal force acting as artificial gravity against the outer hull

Center of Mass

The center of mass is the average position of all the mass that makes up an object

Diagram showing the center of mass location in an irregular object where mass is balanced in all directions

Center of Mass

Objects rotate about their center of mass 

Multi-exposure image of a thrown wrench spinning about its center of mass along a smooth parabolic flight path
Diagram of an irregularly shaped object rotating around its center of mass during projectile motion

Center of Gravity

The center of gravity is the average position of all the weight of an object

For objects on and near Earth surface, center of gravity is the same as center of mass

But it can be different for objects that extend far from the surface of earth. For example The center of gravity of the Sears Tower in Chicago is about 1 mm below its center of mass because the lower stories are pulled a little more strongly by Earth’s gravity than the upper stories.

Photograph of the Willis Tower (Sears Tower) in Chicago illustrating difference between center of mass and center of gravity

Center of Gravity

The center of gravity is the average position of all the weight of an object

Diagram of a meter stick balanced horizontally on a finger at its center of gravity

The weight of the entire stick behaves as if it were concentrated at its center.

Center of Gravity

The center of gravity is the average position of all the weight of an object

The center of gravity can be outside of the object

Diagram of an L-shaped boomerang object showing its center of gravity located in open space outside its physical body

Center of Gravity

The center of gravity is the average position of all the weight of an object

If the center of gravity of an object is above the area of support, the object will remain upright.

Diagram of the Leaning Tower of Pisa showing stability because its vertical center of gravity line falls within its support base

Center of Gravity

The center of gravity is the average position of all the weight of an object

If the center of gravity of an object is above the area of support, the object will remain upright.

Diagram illustrating when an object tips over because its center of gravity extends beyond its support area

Center of Gravity

The center of gravity is the average position of all the weight of an object

Gyroscopes and computer- assisted motors in the self- balancing electric scooter make continual adjustments to keep the combined CGs of Mark, Tenny, and the vehicles above the support base.

Photo of riders on Segway electric scooters maintaining balance by keeping center of gravity over the wheel base

Center of Gravity

The center of gravity is the average position of all the weight of an object

Illustration of a person attempting to bend forward against a wall showing instability when the center of gravity moves outside the feet support area