8-1 Torque

Torque A quantity that measures the ability of a force to rotate an object around some axis.

Greek letter ‘tau’

Net Torque produces rotation.Just like a net force

causes motion/acceleration.

Torque depends upon a force and the length of the

lever arm.

How easily an object rotates depends not only on ‘how much’ force is applied, but also where the force is applied.

Lever ArmThe perpendicular distance from the axis of rotation to a line drawn along the direction of the force.

Torque also depends upon the angle

between a force and the lever arm.

l

= Fd(sin)Metric Units of torque are Newton-meters (N-m)

Torque, like displacement and force,

is a vector quantity.Since torque is a rotational motion it has two directions:

1.Counterclockwise (CCW), which is positive.

2.Clockwise (CW), which is negative.

Ex 1: Sean has a flat and is changing his tire, to remove

the lug nuts he applies a force of 25 N on the tire iron at an

angle of 87o. What is the torque produced if the tire iron

is 0.6 m long?

G: d = 0.6 m, F= 25 N, = 87o

U: = ?

E: = Fd(sin)S: =

S: =

Ex 2: What angle produces a torque of 400 N-m, if the force applied is 505 N at a distance from the axis of rotation of 0.82 m?

G: = 400 N-m, F = 505 N, d =0.82

m

U: = ?

G: = 400 N-m, F = 505 N, d =0.82

m

U: = ?

E:

Fd

sin

G: = 400 N-m, F = 505 N, d =0.82

m

U: = ?

E:

Fd

1sin

)82.0)(505(

400sin : 1S

)82.0)(505(

400sin : 1S

:S

8-2 Rotation and Inertia

Point massWhere all the mass is assumed to be located in one point.

Center of MassThe point at which all the mass of a body can be considered to be concentrated when analyzing translational motion.

Rotational and translational motion can

be combined.We use the the center of mass, as a reference, to analyze its translation

motion.

Center of GravityThe position at which the gravitational force acts on an extended object as if it were a point mass.

Toppling• If the Center of Gravity (CG) is

above the support area, then the object will remain upright.

• If the CG extends outside the support area, the object will topple.

                 

        

Unstable EquilibriumIs when any movement/

displacement of a balanced object lowers the CG.

Stable Equilibrium

•Is when any motion/displacement of a balanced object raises its CG.

Neutral Equilibrium•Is when any motion/displacement of a balanced object neither raises nor lowers its CG.

Moment of Inertia (MOI)

The tendency of a body to rotate freely about a fixed axis to resist a change in rotational motion.

**MOI is the rotational analog of mass.

**Very similar to mass, but MOI is not an intrinsic property of an object.

**It depends upon the object’s mass and

the distribution of mass around the axis

of rotation.

The farther the mass is, on average from the axis of rotation, the greater the

the object’s MOI and the more difficult it is to

rotate the object.

Calculating the MOIPg 285: Table 8-1 has equations/formulas for a few common shapes.

M – mass in kilogramsR – radius in metersl – length in meters

Units for MOI are kg-m2

For an object to be in complete

equilibrium, requires zero net

force and zero net torque.

If the net force is zero the object is in translational

equilibrium. (1st Condition of Equilibrium)

If the net torque is zero, than it is in

rotational equilibrium. (2nd Condition for

Equilibrium)

8-3 Rotational Dynamics

Newton’s 2nd Law for Rotation

F = ma

We know in a rotational system, torque is a function of force, also:

1. MOI replaces the mass.

2. Acceleration is replaced with angular acceleration.

Net torque = MOI x Angular acel.

Inet

Remember that if the net torque is zero, that the object can

still be rotating, just at a constant velocity.

Ex 4: Simon decides to ride the ‘Gravitron’, it has a radius of 3 m, if his mass is 79.4 kg. What is the net torque produced when his angular acceleration is 7 rads/sec2?

G: R = 3 m, M= 79.4 kg, = 7 rads/sec2

U: net = ?

E: net = I

We are going to assume he is a point mass.

MOI, I = MR2

S:net = MR2

S:net =

S:net =

Ex 5: In, 1995, a fully functional pencil with a mass of 24 kg and a length of 2.74 m was made. Suppose this pencil is suspended at its midpoint and a force of 1.8 N is applied perpendicular to its end, causing it to rotate. What is the angular acceleration of the pencil?

G: M = 24kg, F = 1.8 N, l = 2.74 m

U: = ?

E: = I = / IWe need to find the values for Torque and MOI.

Since the midpoint is the pivot point the lever arm is ½ l. Also, the force acts perpendicular, so the angle is 90 degrees.

= Fd(sin = ½Fl (sin)

= ½(1.8)(2.74)sin90

=

We use the formula for MOI on pg 285 for a thin rod rotating around its midpoint.

I = 1/12 Ml 2 I =I =

S: =

S: =

Angular Momentum (L)

The product of a rotating object’s moment of

inertia and the angular speed about the same

axis.

L = IL = angular momentum

I = Moment of Inertia

= angular speed

Angular Momentum Units

Kg-m2/s

Law of Conservation of Momentum

When the net external torque acting on an object or objects

is zero, the angular momentum does not change.

Li = Lf

Ex 6: A figure skater jumps into the air to perform a twisting maneuver. When she first jumps her moment of inertia is 86 kg-m2. While she’s in the air she brings her arms in and decreases her moment of inertia to 77 kg-m2. If her initial angular speed was 2 rads/sec what is her final angular speed, if momentum is conserved?

G: i = 2 rads/sec, Ii = 86 kg-m2 , If = 77 kg-m2

U: f = ?Since momentum is conserved:

E: Li = Lf

Iii = Iff

f = Iii / If

S: f =

S: f =

Rotational Kinetic Energy (KERot)

Energy of an object due to its rotational motion.

Calculating Rotational Kinetic Energy

KErot = ½ I2

KErot = rotational KE I = moment of inertia = angular speed

EX 8: What is the MOI for a 0.75 kg top

spinning at an angular speed of 1.75 rads/sec, if the Rotational KE is

12 Joules?

G: M = 0.75 kg, KErot = 12 J, = 1.75 rads/sec

U: I = ?

E: KErot = ½ I2

I = 2 KErot /2

S: I =

S: I =

Remember Mechanical Energy may be conserved. ME is the sum of all types of KE and PE.

ME = KEtrans + KErot + PE

ME = ½ mv2 + ½ I2 + mgh

If ME is conserved:

MEinitial = MEfinal

If ME is conserved:

MEinitial = MEfinal

ffrotftrans

iirotitrans

PEKEKE

PEKEKE

,,

,,

fff

iii

mghImv

mghImv

22

122

1

22

122

1