Chapter 8 Rotational Motion

8.1 Describing Rotational Motion When an object spins, It is said to undergo Rotational motion.

When measuring rotational Motion, we will use radians. A radian is an angle whose Arc length is equal to its radius, Which is approximately Equal to 57.3°

s r θ = θ = the angle r = the radius s = arc length

We now have a conversion Between radians and degrees. π 180° θ (rad) = θ (deg)

Ben, riding on a carousel, Problem... Ben, riding on a carousel, Travels through an arc length of 11.5 m. If the carousel has a Radius of 8 m, what is the angular Displacement? What degree is this? 2.88 rad 165°

Angular speed describes Rotates about an axis, usually The rate at which a body Rotates about an axis, usually Expressed in rad/s. Δθ Δt ω =

Ben now spins on a stool, if he Turns clockwise through 10π rad, Problem... Ben now spins on a stool, if he Turns clockwise through 10π rad, During a 10 s interval, what is The average angular speed Of his feet? 3.14 rad/s

Angular acceleration is the time Rate of change of angular speed, Usually expressed in radians Per second per second. Δω Δt α =

Now Ben rotates on the stool With an initial angular speed of Problem... Now Ben rotates on the stool With an initial angular speed of 21.5 rad/s. The stool accelerates, And after 3.5s, the speed is 28 rad/s. What is the average Angular acceleration? 1.9 rad/s/s

Angular Kinematics ωf = ωi + αΔt vf = vi + aΔt Δθ = ωiΔt + ½α(Δt)2 Rotational Motion Linear ωf = ωi + αΔt vf = vi + aΔt Δθ = ωiΔt + ½α(Δt)2 Δx = viΔt + ½a(Δt)2 ωf2 = ωi2 + 2α(Δθ) vf2 = vi2 + 2aΔx

The wheel on an upside-down bike Rotates with a constant angular Problem... The wheel on an upside-down bike Rotates with a constant angular Acceleration of 3.5 rad/s/s. If the Initial angular speed of the wheel Is 2 rad/s, through what angular Displacement does the wheel Rotate in 2 sec? 11 rad

Torque is a quantity that measures The ability of a force to rotate 8.2 Rotational Dynamics Torque is a quantity that measures The ability of a force to rotate An object around some axis

Torque is a scalar quantity. Torque has the SI unit of N*m τ = Fd(sinθ)

Torque is positive or negative Depending on the direction The force tends to rotate An object. Torque that is clockwise is Defined as negative. Thus torques that produce a Counterclockwise rotation are Defined to be positive.

Find the torque produced by a 3 N force applied at an angle Problem... Find the torque produced by a 3 N force applied at an angle Of 60° to a door 0.25m from The hinge. 0.65 N m

In order to calculate the moment Of inertia, you have to use the Formulas provided in the book For the many different shapes.

Formula for moment of inertia Thin hoop about symmetry of object MR2 Shape Formula for moment of inertia Thin hoop about symmetry of object MR2 Thin hoop about diameter 1/2 MR2 Point of mass about axis Disk about axis Rod about axis through center 1/12 ML2 Rod about axis through end 1/3 ML2 Solid sphere about diameter 2/5 MR2 Thin sphere shell about diameter 2/3 MR2

A 5m horizontal beam weighing 315N is attached to a wall so Problem... A 5m horizontal beam weighing 315N is attached to a wall so That it rotates. Its far end is Supported by a cable at an angle Of 53°, and a 545N is standing 1.5m from the wall. Find the tension And the force on the beam by the Wall, R. Tension = 403 N R = 590N

Newton’s 2nd Law for Rotating objects. τ = Iα

Stone leaves the catapult with an Acceleration of 100 m/s2, what Problem... A catapult propels a 0.150 kg Stone. The length of the Catapult arm is 0.350m. If the Stone leaves the catapult with an Acceleration of 100 m/s2, what Is the torque exerted on the stone. 5.25 Nm

The center of mass of an object Is the point at which all the 8.3 Equilibrium The center of mass of an object Is the point at which all the Mass of the body can be Considered to be concentrated When analyzing Translational motion.

Rotational and translational Motion can be combined. The moment of inertia is the Rotational analog of mass. Equilibrium requires zero net force, And zero net torque.

On 2 saw horses. Sawhorse A is 0.6 m from one end of the ladder Problem... A 5.8 kg ladder, 1.8 m long, rests On 2 saw horses. Sawhorse A is 0.6 m from one end of the ladder And sawhorse B is 0.15 m from the Other end of the ladder. What force Does each sawhorse exert on the Ladder? Fb = 16 N Fa = 41 N

The centrifugal force is an Apparent force. NOT A REAL FORCE! It is not a real force because there Is no physical outward push.

The Coriolis Force is also a FAKE FORCE! This is because it is only apparent To rotating observer. This is very apparent on Earth However. It is the reason for The direction of the winds.

THE END