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**Dynamics of Rotational Motion**

Chapter 10 Dynamics of Rotational Motion

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Goals for Chapter 10 To learn what is meant by torque To see how torque affects rotational motion To analyze the motion of a body that rotates as it moves through space To use work and power to solve problems for rotating bodies To study angular momentum and how it changes with time

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Loosen a bolt Which of the three equal-magnitude forces in the figure is most likely to loosen the bolt? Fa? Fb? Fc?

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Loosen a bolt Which of the three equal-magnitude forces in the figure is most likely to loosen the bolt? Fb!

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**Let O be the point around which a solid body will be rotated.**

Torque Let O be the point around which a solid body will be rotated. O

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**Let O be the point around which a solid body will be rotated.**

Torque Line of action Let O be the point around which a solid body will be rotated. Apply an external force F on the body at some point. The line of action of a force is the line along which the force vector lies. O F applied

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**Let O be the point around which a body will be rotated.**

Torque Line of action Let O be the point around which a body will be rotated. The lever arm (or moment arm) for a force is The perpendicular distance from O to the line of action of the force L O F applied

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Torque Line of action The torque of a force with respect to O is the product of force and its lever arm t = F L Greek Letter “Tau” = “torque” Larger torque if Larger Force applied Greater Distance from O L O F applied

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**Torque t = F L Larger torque if Larger Force applied**

Greater Distance from O

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Torque Line of action The torque of a force with respect to O is the product of force and its lever arm t = F L Units: Newton-Meters But wait, isn’t Nm = Joule?? L O F applied

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**Torque Torque Units: Newton-Meters Use N-m or m-N**

OR...Ft-Lbs or lb - feet Not… Lb/ft Ft/lb Newtons/meter Meters/Newton

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Torque Torque is a ROTATIONAL VECTOR! Directions: “Counterclockwise” (+) “Clockwise” (-) “z-axis”

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Torque Torque is a ROTATIONAL VECTOR! Directions: “Counterclockwise” “Clockwise”

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Torque Torque is a ROTATIONAL VECTOR! Directions: “Counterclockwise” (+) “Clockwise” (-) “z-axis”

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Torque Torque is a ROTATIONAL VECTOR! But if r = 0, no torque!

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**Torque can be expressed as a vector using the vector cross product:**

Torque as a vector Torque can be expressed as a vector using the vector cross product: t= r x F Right Hand Rule for direction of torque.

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**Torque can be expressed as a vector using the vector cross product:**

Torque as a vector Torque can be expressed as a vector using the vector cross product: t= r x F Where r = vector from axis of rotation to point of application of Force q = angle between r and F Magnitude: t = r F sinq Line of action Lever arm O r q F applied

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**Torque can be evaluated two ways:**

Torque as a vector Line of action Torque can be evaluated two ways: t= r x F t= r * (Fsin q) remember sin q = sin (180-q) Lever arm O r q F applied Fsinq

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**Torque can be evaluated two ways:**

Torque as a vector Line of action Torque can be evaluated two ways: t= r x F t= (rsin q) * F The lever arm L = r sin q Lever arm r sin q O q r F applied

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**Torque as a vector Torque is zero three ways:**

Line of action Torque is zero three ways: t= r x F t= (rsin q) * F t= r * (Fsin q) If q is zero (F acts along r) If r is zero (f acts at axis) If net F is zero (other forces!) Lever arm r sin q O q r F applied

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Applying a torque Find t if F = 900 N and q = 19 degrees

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Applying a torque Find t if F = 900 N and q = 19 degrees

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**Torque and angular acceleration for a rigid body**

The rotational analog of Newton’s second law for a rigid body is z =Iz. What is a for this example?

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**Torque and angular acceleration for a rigid body**

The rotational analog of Newton’s second law for a rigid body is z =Iz. t = Ia and t = Fr so Ia = Fr a = Fr/I and a = r a

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**Another unwinding cable – Example 10.3**

What is acceleration of block and tension in cable as it falls?

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**Rigid body rotation about a moving axis**

Motion of rigid body is a combination of translational motion of center of mass and rotation about center of mass Kinetic energy of a rotating & translating rigid body is KE = 1/2 Mvcm2 + 1/2 Icm2.

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**Rolling without slipping**

The condition for rolling without slipping is vcm = R.

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A yo-yo What is the speed of the center of the yo-yo after falling a distance h? Is this less or greater than an object of mass M that doesn’t rotate as it falls?

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**The race of the rolling bodies**

Which one wins?? WHY???

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**Acceleration of a yo-yo**

We have translation and rotation, so we use Newton’s second law for the acceleration of the center of mass and the rotational analog of Newton’s second law for the angular acceleration about the center of mass. What is a and T for the yo-yo?

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**Acceleration of a rolling sphere**

Use Newton’s second law for the motion of the center of mass and the rotation about the center of mass. What are acceleration & magnitude of friction on ball?

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**Work and power in rotational motion**

Tangential Force over angle does work Total work done on a body by external torque is equal to the change in rotational kinetic energy of the body Power due to a torque is P = zz

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**Work and power in rotational motion**

Example 10.8: Calculating Power from Torque Electric Motor provide 10-Nm torque on grindstone, with I = 2.0 kg-m2 about its shaft. Starting from rest, find work W down by motor in 8 seconds and KE at that time. What is the average power?

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**Work and power in rotational motion**

Analogy to 10.8: Calculating Power from Force Electric Motor provide 10-N force on block, with m = 2.0 kg. Starting from rest, find work W down by motor in 8 seconds and KE at that time. What is the average power? W = F x d = 10N x distance travelled Distance = v0t + ½ at2 = ½ at2 F = ma => a = F/m = 5 m/sec/sec => distance = 160 m Work = 1600 J; Avg. Power = W/t = 200 W

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**Angular momentum Units = kg m2/sec (“radians” are implied!)**

The angular momentum of a rigid body rotating about a symmetry axis is parallel to the angular velocity and is given by L = I. Units = kg m2/sec (“radians” are implied!) Direction = along RHR vector of angular velocity.

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**Angular momentum For any system of particles = dL/dt.**

For a rigid body rotating about the z-axis z = Iz. Ex 10.9: Turbine with I = 2.5 kgm2; w = (40 rad/s3)t2 What is L(t) & t = 3.0 seconds; what is t(t)?

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**Conservation of angular momentum**

When the net external torque acting on a system is zero, the total angular momentum of the system is constant (conserved). Follow Example using Figure below.

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**A rotational “collision”**

Follow Example using Figure at the right.

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**Angular momentum in a crime bust**

A bullet (mass = 400 m/s) hits a door (1.00 m wide mass = 15 kg) causing it to swing. What is w of the door?

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**Angular momentum in a crime bust**

A bullet (mass = 400 m/s) hits a door (1.00 m wide mass = 15 kg) causing it to swing. What is w of the door? Lintial = mbulletvbulletlbullet Lfinal = Iw = (I door + I bullet) w Idoor = Md2/3 I bullet = ml2 Linitial = Lfinal so mvl = Iw Solve for w and KE final

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**Gyroscopes and precession**

For a gyroscope, the axis of rotation changes direction. The motion of this axis is called precession.

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**Gyroscopes and precession**

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**Gyroscopes and precession**

For a gyroscope, the axis of rotation changes direction. The motion of this axis is called precession.

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A rotating flywheel Magnitude of angular momentum constant, but its direction changes continuously.

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**A precessing gyroscopic**

Example 10.13

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