Chapter 10Rotational motionand Energy

Rotational MotionUp until now we have been looking at the kinematics and dynamics of translational motion that is, motion without rotation. Now we will widen our view of the natural world to include objects that both rotate and translate.

We will develop descriptions (equations) that describe rotational motion

Now we can look at motion of bicycle wheels and even more!

II. Rotation with constant angular acceleration

III. Relation between linear and angular variables- Position, speed, accelerationRotational variables - Angular position, displacement, velocity, accelerationIV. Kinetic energy of rotation

V. Rotational inertia

VI. Torque

VII. Newtons second law for rotation

VIII. Work and rotational kinetic energy

Rotational kinematicsIn the kinematics of rotation we encounter new kinematic quantitiesAngular displacementqAngular speedwAngular accelerationaRotational InertiaITorquetAll these quantities are defined relative to an axis of rotation

Angular displacementMeasured in radians or degreesThere is no dimensionDq = qf qi CWAxis of rotationDqqiqf

Note: we do not reset to zero with each complete rotation of the reference line about the rotation axis. 2 turns =4Translation: bodys movement described by x(t). Rotation: bodys movement given by (t) = angular position of the bodys reference line as function of time.Angular displacement: bodys rotation about its axis changing the angular position from 1 to 2.Clockwise rotation negativeCounterclockwise rotation positive

Angular displacement and arc lengthArc length depends on the distance it is measured away from the axis of rotation

Angular SpeedAngular speed is the rate of change of angular position

We can also define the instantaneous angular speed

Average angular velocity and tangential speedRecall that speed is distance divided by time elapsedTangential speed is arc length divided by time elapsed

And because we can write

Average Angular AccelerationRate of change of angular velocity

Instantaneous angular acceleration

Angular acceleration and tangential accelerationWe can find a link between tangential acceleration at and angular acceleration

So

Centripetal accelerationWe have that

But we also know thatSo we can also say

Example: RotationA dryer rotates at 120 rpm. What distance do your clothes travel during one half hour of drying time in a 70 cm diameter dryer? What angle is swept out?Distance: s = Dq r and w = Dq/Dt so s = wDtrs = 120 /min x 0.5 h x 60 min/h x 0.35 m = 1.3 kmAngle: Dq = w Dt = 120 r/min x 0.5 x 60 min = 120x2pr /min x 0.5 h x 60 min/h = 2.3 x 104 r

Rotational motion with constant angular accelerationWe will consider cases where a is constantDefinitions of rotational and translational quantities look similarThe kinematic equations describing rotational motion also look similarEach of the translational kinematic equations has a rotational analogue

Rotational and Translational Kinematic Equations

Constant a motionWhat is the angular acceleration of a cars wheels (radius 25 cm) when a car accelerates from 2 m/s to 5 m/s in 8 seconds?

Example: Centripetal AccelerationA 1000 kg car goes around a bend that has a radius of 100 m, travelling at 50 km/h. What is the centripetal force? What keeps the car on the bend? [What keeps the skater in the arc?]Friction keeps the car and skater on the bend

Car rounding a bendFrictional force of road on tires supplies centripetal forceIf ms between road and tires is lowered then frictional force may not be enough to provide centripetal forcecar will slideLocking wheels makes things worse asmk < msBanking of roads at corners reduces the risk of skidding

Car rounding a bendHorizontal component of the normal force of the road on the car can provide the centripetal forceIf

then no friction is requiredFgNNcosqNsinqq

Rotational DynamicsEasier to move door at A than at B using the same force F

More torque is exerted at A than at BABhinge

TorqueTorque is the rotational analogue of ForceTorque, t, is defined to be

Where F is the force applied tangent to the rotation and r is the distance from the axis of rotation

rFt = Fr

TorqueA general definition of torque is

Units of torque are NmSign convention used with torqueTorque is positive if object tends to rotate CCWTorque is negative if object tends to rotate CW

qt = Fsinq r

Condition for EquilibriumWe know that if an object is in (translational) equilibrium then it does not accelerate. We can say that SF = 0 An object in rotational equilibrium does not change its rotational speed. In this case we can say that there is no net torque or in other words that:St = 0

An unbalanced torque (t) gives rise to an angular acceleration (a)We can find an expression analogous to F = ma that relates t and aWe can see thatFt = matand Ftr = matr = mr2a (since at = ra)Therefore

Torque and angular accelerationmrFtt = mr2a

Torque and Angular AccelerationAngular acceleration is directly proportional to the net torque, but the constant of proportionality has to do with both the mass of the object and the distance of the object from the axis of rotation in this case the constant is mr2This constant is called the moment of inertia. Its symbol is I, and its units are kgm2I depends on the arrangement of the rotating system. It might be different when the same mass is rotating about a different axis

Newtons Second Law for Rotation Now we have

Where I is a constant related to the distribution of mass in the rotating systemThis is a new version of Newtons second law that applies to rotationt = Ia

Angular Acceleration and IThe angular acceleration reached by a rotating object depends on, M, r, (their distribution) and T

When objects are rolling under the influence of gravity, only the mass distribution and the radius are importantT

Moments of Inertia for Rotating ObjectsThe total torque on a rotating system is the sum of the torques acting on all particles of the system about the axis of rotation and since a is the same for all particles:I Smr2 = m1r12+ m2r22+ m3r32+

Axis of rotation

Continuous Objects To calculate the moment of inertia for continuous objects, we imagine the object to consist of a continuum of very small mass elements dm. Thus the finite sum mi r2i becomes the integral

Moment of Inertia of a Uniform Rod LLets find the moment of inertia of a uniform rod of length L and mass M about an axis perpendicular to the rod and through one end. Assume that the rod has negligible thickness.

Moment of Inertia of a Uniform Rod We choose a mass element dm at a distance x from the axes. The mass per unit length (linear mass density) is = M / L

Moment of Inertia of a Uniform Rod dm = dx

Example:Moment of Inertia of a DumbbellA dumbbell consist of point masses 2kg and 1kg attached by a rigid massless rod of length 0.6m. Calculate the rotational inertia of the dumbbell (a) about the axis going through the center of the mass and (b) going through the 2kg mass.

Example:Moment of Inertia of a Dumbbell

Example:Moment of Inertia of a Dumbbell

Moment of Inertia of a Uniform Hoop RdmAll mass of the hoop M is at distance r = R from the axis

Moment of Inertia of a Uniform Disc RdrEach mass element is a hoop of radius r and thickness dr. Mass per unit area = M / A = M /R2

rWe expect that I will be smaller than MR2 since the mass is uniformly distributed from r = 0 to r = R rather than being concentrated at r=R as it is in the hoop.

Moment of Inertia of a Uniform Disc Rdr r

Moments of inertia I for Different Mass Arrangements

Moments of inertia I for Different Mass Arrangements

Units: NmTangential component, Ft: does cause rotation pulling a door perpendicular to its plane. Ft= F sin Radial component, Fr : does not cause rotation pulling a door parallel to doors plane.TorqueTorque: Twist Turning action of force F .

r : Moment arm of Fr : Moment arm of FtSign: Torque >0 if body rotates counterclockwise. Torque

Newtons second law for rotationProof:Particle can move only along the circular path only the tangential component of the force Ft (tangent to the circular path) can accelerate the particle along the path.

Kinetic energy of rotationReminder: Angular velocity, is the same for all particles within the rotating body.

Linear velocity, v of a particle within the rigid body depends on the particles distance to the rotation axis (r).Moment of Inertia

Rotational Kinetic EnergyWe must rewrite our statements of conservation of mechanical energy to include KErMust now allow that (in general):

mv2+mgh+ Iw2 = constantCould also add in e.g. spring PE

VII. Work and Rotational kinetic energyTranslationRotationWork-kinetic energy TheoremWork, rotation about fixed axisWork, constant torquePower, rotation about fixed axisProof:

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