Physics 101

Prof. Ekey

Chapter 12

Rotation of a Rigid Body Torque, Inertia, Rot. Energy, Rot. Ang. Mom.

“In this chapter, you will learn to understand and apply the physics of rotation”

Rotations: Redux

Please look at chapter 4 (again)

Rigid body – extended object whose size and shape do not

change as it moves. Ex: Bicycle wheel

Speed at a radius. Velocity continuously changes directions

Acceleration at a radius continuously changes directions

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ac = ar =v2

r= rω2 Points

radially inward

Angular Velocity Units: rad/s

Angular Accel. Units: rad/s2

Tangential Velocity

Tangential Acceleration

Centripetal Acceleration

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ω =dθdt

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α =dωdt

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vt = rω

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at = rα

Rate at which the angle is changing

Rate at which angular velocity is changing

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θ f =θi +ωiΔt + 12α(Δt)

2

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ω f =ωi +α(Δt)

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ω f2 =ωi

2 + 2αΔθ

Rotational/Angular Kinematic Equations Signs of angular velocity and acceleration Counter clockwise (ccw) is positive Clockwise (cs) is negative

Can also use the right-hand-rule (into/out of the page)

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xcm =1M

mixii∑ =

m1x1 +m2x2 +m3x3 + ....m1 +m2 +m3

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ycm =1M

miyii∑ =

m1y1 +m2y2 +m3y3 + ....m1 +m2 +m3

Center of Mass - “balancing point”

The ccw, cw convention is what the book uses.

\“point at which all of the mass of an object or a system may be

considered to be concentrated” “mass-weighted center of the object”

Describe the motion of a many particle system w.r.t. the C of M.

Why? Because it is easier, and the same physics apply.

A “free” object will rotate around its center of mass.

Model object/system as if it were concentrated at multiple points

Two masses, 1.0 kg and 2.0kg, are located as shown. What is the location of their C of M?

If the system rotates with w=5.0 rad/s, what are the tangential speeds of each mass?

10 0 5 m1kg m2kg

x [m]

C of M, catch the excitement. The center of mass of a system consisting of two 1.0 kg particles is located at the origin. If one of the particles is at (0, 0.50 m) where is the other? If the masses are moved so their center of mass is located at (0.25 m, 0.50 m) can you tell where the particles are located?

(a) Yes (b) No (c) Maybe (d) Need more information

Rotational Kinetic Energy: Units: Nm or J

What’s the sign of rotational KE?

Moment of Inertia, I Rot. equiv. of mass Units: kg m2

A 2.0 kg ball is spun on a 1.0 m long massless string, at a speed of 4.0 m/s. What is the moment of intertia of the ball? What is the rotational kinetic energy of the ball?

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Krot = 12 Iω

2

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I = mii∑ ri

2

Check the units

Moment of inertia depends on the axis of rotation,

and the distance of the masses from the axis.

What happens to I & Krot if you double the radius (same w)? What about the mass?

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I = mr2

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I = miri2

i∑

Notice they all have the same form. # times mass times length squared Moment of inertia…

Point Particle

Multiple Point

Particles

You will have this for the final

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Isys = I1 + I2 + I3 + ...If you know both objects moment

of inertia, then just add them up.

You have a disc of mass 20 kg and radius 0.25 m with another mass of 5.0 kg and a radius of 0.10 m atop and centered over their center of mass. What is the rotational inertia about their center of mass?

Parallel axis-theorem. Moment of inertia about an axis that is

parallel to one through its center of mass a distance, d, from c. of m.

Calculate I for a 1.0 kg meter stick

oscillating about its 0.25 m mark

I = ICM + Md 2

Moment of inertia for combined/unknown system

pivot point

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τ ≡ rF sinφ

Torque (t) – can cause rotations Units: Nm (not J) Rotational equivalent of force. “effectiveness of force in causing rot.”

Depends on the magnitude of the applied force F

The distance, r, from the point of application to the pivot

The angle f between the force and radius vector

If F = 2.0 N, r = 0.25 m and f = 100º

What is t?

Direction of rotation?

Vector How does a seesaw work? What’s a lever good for anyway?

magnitude

If r=1.0 m and F =1.0N, and the angles

between each force vector are 45º (or 90º),

what is the mag. and sign of each torque?

CCW Torques are positve

CW Torques are negative

What is the torque if you push/pull in the direction of the radial line?

Torques and Signs.

1 2

3

4

5 6

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τ = rFt

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τ = dF = (r sinφ)F

Torques and moment arms

Torque due to the tangential

component of the force

This is what we’ve been doing.

Alternatively the distance (d=r sin f) from the pivot point to the line

of action (the line along which the force acts)

d is also called the moment/lever arm

Angle between d and F is 90º

Go back to previous problem

and draw in some lever arms

Mass of door: 20 kg Width of door: 1.0 m

Which door do you buy?

Newton’s II law for rotations

“An object’s angular acceleration is equal to the torque exerted on it divided by its moment of inertia (rotational mass). The angular acceleration is in the same direction as the torque.”

Joe makes doors with the door knob located in the middle of the door. Pete makes doors that have the same size and mass as Joe’s doors, except that the door knob is close to the side of the door opposite of the hinges. If a constant 5.0 N Force is applied to each knob (perpendicular to the door), what torque do you apply to each door? Calculate the moment of inertia for each door. What is the angular acceleration of each door?

τ net =

τ i∑ = I α

DVD player revisited A 12.0 cm diameter and 20 g DVD accelerates from rest to its operational speed of 500 rpm in 3.50 seconds. Calculate the angular acceleration of the DVD.

Calculate the moment of inertia of the disc

Determine the torque applied to the DVD during start-up What angle does the DVD turn through during start-up?

Questions You are using a wrench and trying to loosen a rusty nut. Which of the arrangements shown is creates the most torque? Two masses are connected via rigid, massless rod and the forces are applied as shown. What is the angular acceleration of the system? (a) 0 rad/s2 (b) 2 rad/s2 (c) 4 rad/ s2 (d) 8 rad/s2

(a) (b) (c) (d)

Falling meter stick. You release a 0.50 kg meter-stick from a wall. Calculate the torque and angular acceleration when q = 15º Draw a picture

What happens to a and t as q increases? Do you want to solve this with kinematics?

Use Energy Conservation to find the angular velocity of the stick right before it hits the ground.

Wal

l

q

Pivot point

Static Equilibrium (balanced and/or stable)

Net force and Net torque equal zero

translational and rotational equilibrium condition

Unbalanced forces produce translational acceleration

Unbalanced torques produce rotational acceleration

Stable Equilibrium – Any small displacement results in restoring force or torque, which tends to return the object to its original equilibrium position.

Unstable Equilibrium – Any small displacement results in a torque,

which tends to rotate the object farther from its equilibrium position.

As long as the center of mass (CoM) still lies inside and above the

objects original base of support, it will be stable.

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F net =

F i = 0

τ net = τ i∑∑ = 0

CoM CoM

Question A 1.0 kg rock is suspended by a mass-less string from one end of a

1.0 m long measuring stick. What is the weight of the measuring stick

if it is balanced by a support force at the 0.25-m mark?

(a) 0.25 kg

(b) 0.5 kg

(c) 1.0 kg

(d) 2.0 kg

(e) 4.0 kg

Draw a free-body diagram of all forces acting on system. (Normal Force?) What’s the net force? What’s the net torque?

For this problem

Pivot

The Ladder.

A 3.0 m long ladder leans agains a

frictionless wall at an angle of 60º

What is the static friction force

necessary to keep it from slipping?

Net Force in x direction?

Net Force in y direction?

Net Torque? Calculate fs

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d1 = 12 L cos60º

d2 = L sin60º

How would you calculate µs?

Ladder mass = 100 kg

Translation and Rotation Motion (how we move) Rigid Body – object where distance between particles are fixed

Translational motion – Linear motion

Pure Translation- particles have same instantaneous velocity

Rotational Motion – Rotational motion about a fixed axis

Pure Rotation- particles have same instantaneous angular velocity

Rolling – Combination of both (Curve Ball, Slice, Fade, “English”)

Axis of Rotation

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Krolling = 12 Icmω

2 + 12 Mvcm

2

Krolling = 12 Icm

vcmr

⎛

⎝⎜

⎞

⎠⎟

2

+ 12 mvcm

2 = 12 Icmω

2 + 12 m(rω)2

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vcm = rω

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s = rθ vcm = rω acm = rα

Rolling without slipping (we want static friction)

Can relate Translation and Rotation motion

Arc Length Velocity Acceleration

Kinetic Energy of a rolling body A rolling body has both translational and rotational kinetic energy

Total KE = Rotational KE + Translational KE of CoM

Using the no slipping condition you can write this as

For a ball/cylinder - Center of mass over point of contact

A car that is traveling up a hill a patch of ice that causes its tires to spin. Which statements is correct? (a) vcm > rw (b) rw > vcm (c) vcm = rw (d) Need more information

A hoop and a solid disc of the same radius and mass roll from rest down an incline. Which reaches the bottom of the incline first?

(a) The hoop (b) The solid disc (c) Both reach the bottom at the same time. (d) Need more information.

Which has a larger moment of intertia? Largest vcm at bottom of incline?

Questions

Difference between slipping and sliding?

Total Mechanical Energy

Conservation of total ME – for no slip rolling systems

Look back to ramp example.

Determine vCM at the bottom of the ramp. Could ask for w as well…

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Emech = K +U

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K f +U f = Ki +Ui = ETot (CONSTANT )

Same as b4, except we have rotational KE

h =0.50 m m

r

w

vCM

A 0.10 kg hoop rolls from rest down an incline. The hoop has a radius of 0.10 m and starts from a height of 0.50 m.

What is the total mechanical energy before release & at the bottom of the ramp?

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L = mr2 vtr

"

# $

%

& ' = mrvt

Angular Momentum, L Units: kg m2/s

Points in the same direction as w (RHR)

Newton’s II law

in terms of ∆L.

On the Price is Right (Barker > Carey) you need to make it into the showcase shown down by spinning the big wheel. The solid wheel has a mass of 200 kg and a radius of 2.0 m and you apply a constant 50 Nm torque for 1.0 s to move the wheel. What is the change in angular momentum of the wheel? What is the final angular velocity of the wheel?

L = I

ω

Vector

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τ net = I α = I Δ

ω Δt

=Δ(I ω )Δt

=Δ L Δt

=d L

dt

For a point particle

Question

An ice skater spins about a vertical axis through her body with her arms held out. As she draws her arms inwards, her angular velocity

(a) Increases (b) Decreases (c) Remains the same (d) Is this the Olympics?

How is the mass changing? How is the inertia changing? Is there a net torque?

τ net =

ΔLΔt

= ZERO

ΔL =Lf −

Li = 0

Lf =

Li

Lf∑ =

Li∑   recall   L = Iω

Conservation of angular momentum (Ang mom con)

“The angular momentum of an isolated system (net torque = 0) is

conserved. The total initial and final angular momentum are equal”

A 100 g bullet is traveling at 1000 m/s strikes the edge a 1.0 m diameter spinning disc spinning about its center (I = 10 kg m2). If after the collision the bullet and disc are at rest, what is the angular speed of the disc prior to the collision?

Notice to change L either change the

angular velocity or the moment of inertia

Look back to ice-skater. Ang mom con

Keep track of total “L” before and after

Dropping a record on a turntable.

A 2.0 kg free-turning turntable (no external torques) of diameter 16 in. (r = 0.20 m) spins with a constant angular velocity of 331/3 rpm (3.5 rad/s). What happens to the angular velocity of the system if you drop a 1.0 kg record diameter of 12 in. (r = 0.15 m) onto the turntable?

BEFORE Moment of Inertia Angular Velocity Angular Momentum

AFTER Moment of Inertia Angular Velocity Angular Momentum

Take a guess.

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A

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B

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A × B = ABsinα

Cross Product Another way to multiply 2 vectors

Given vectors and and

the angle a between them.

The cross product (vector product) of the two vectors is

Where A and B are the magnitude of the vectors. Points in the direction given by the right hand rule (RHR).

Value is a vector

Thumb in direction of vector “A”. Pointer finger in direction of B. Middle finger in the dirction of the Cross Product of A and B

Point your fingers in the direction of A and curl your fingers towards B. Your thumb points in the direction of the cross product of A and B.

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τ = r × F = rF sinφ

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L = r × p = rpsinβ

A×B = AyBz − AzBy( ) i + AzBx − AxBz( ) j + AxBy − AyBx( ) k

L = r × p = ry pz − rz py( ) i + rz px − rx pz( ) j + rx py − ry px( ) k

L = (−10 ⋅0)− (0 ⋅5)( ) i + (0 ⋅5)− (0 ⋅5)( ) j + (0 ⋅5)− (−10 ⋅5)( ) k   (Most  comp. are zero)L = −50 kgm / s k   

Cross Products, Torques and Angular Momentum Torque in terms of cross product.

Angular Momentum of a particle

Cross product in terms of components

A particle is traveling with momentum

when at a location of what is the angular momentum of

the particle?

Direction given by RHR

r = −10 j m

p = (5i + 5 j) kgm s

Note: I will not ask you this version of cross product on final. L = r F sin angle = yes (or maybe)

Questions

A person spins a ball on a string in a horizontal circle. At the point indicated the ball is given a sharp blow in the x-direction. This causes a change in the angular momentum ∆L in the

(a) x direction (b) y direction (c) z direction

Ans (c). If you apply a force at a distance, you apply a torque to the ball. This increases the angular velocity (w), which in turn increases the angular momentum (I w). By the RHR the angular momentum points perpendicular to the plane of motion, and post hit the angular momentum increase is in the z-direction as well.

L

v F

x

z y

τ net =

ΔLΔt

= rF sinφ