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**Rotational Motion and Equilibrium**

Ch. 8 Rotational Motion and Equilibrium

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**Ch. 8 Overview Rolling Motion Torque Rotational Equilibrium**

Rotational Dynamics Rotational Kinetic Energy Conservation of Angular Momentum

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Tangential Variables To describe the connection between linear and rotational motion, we use tangential variables. Arc length Tangential speed Tangential acceleration

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Arc Length Δs = rΔθ Δθ r Δs

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Tangential Speed Δθ r Δs vt but ω=Δθ/Δt so

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**Ex. The turntable turns at 33 1/3 rpm**

Ex. The turntable turns at 33 1/3 rpm. Find the angular velocity of the motor if the radius of the small pulley is 1.27 cm.

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**Gear Ratio v is the same for both wheels r1ω1 = r2ω2 ω2 = r1/r2 ω1**

γ = r1/r2 is called the gear ratio v r1 r2

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**½ the ladybug’s twice the ladybug’s equal to the ladybug’s**

A ladybug sits at the outer edge of a merry go-round, and a gentleman bug sits halfway between her and the axis of rotation. The merry-go-round makes a complete revolution once each second. The gentleman bug’s angular speed is ½ the ladybug’s twice the ladybug’s equal to the ladybug’s cannot be determined 1 2 3 4 5

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**½ the ladybug’s twice the ladybug’s equal to the ladybug’s**

A ladybug sits at the outer edge of a merry go-round, and a gentleman bug sits halfway between her and the axis of rotation. The merry-go-round makes a complete revolution once each second. The gentleman bug’s tangential speed is ½ the ladybug’s twice the ladybug’s equal to the ladybug’s cannot be determined 1 2 3 4 5

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**Tangential Acceleration**

If the tangential speed changes, we can define a tangential acceleration

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Rolling Motion If a car rolls without slipping, then the distance the car travels = the arc length turned by the wheel d = s = rθ The condition for rolling w/o slipping can then be expressed as v = rω

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**How does the instantaneous velocity of the two points compare?**

vA = vB vA > vB vA < vB Cannot be determined B 1 2 3 4 5

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Rolling w/o Slipping The instantaneous velocity at the point of contact is 0 A perfect wheel has no kinetic friction v = 2 rω v = rω v = 0

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**Ex. A car starts from rest and accelerates to 15 m/s in a time of 5**

Ex. A car starts from rest and accelerates to 15 m/s in a time of 5.0 s. a) Sketch the situation. b) Find the car’s average acceleration during the 5.0 s. c) Find the angular velocity of the tires at t = 5.0 s if their diameter is 16”. d) Find the angular acceleration of the wheel during the 5.0 s.

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**Torque A torque produces an angular acceleration**

Torque = τ = rF sinθ = Fperpendicular ∙r If the torque produces a ccw rotation +, cw - F r θ

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Ex. A force of 10.0 N is applied along the tangent of a wheel of radius .50 m. a) sketch the situation. b) Find the net torque on the wheel.

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**Ex. Find the net force and torque for each situation shown below.**

r = .25 m r = .25 m 5.0 N 5.0 N

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How do the magnitudes of the torques produced by identical forces for the two situations shown compare? A B TA > TB TA = TB TA < TB Cannot be determined 1 2 3 4 5

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How do the magnitudes of the torques produced by identical forces for the two situations shown compare? B A TA > TB TA = TB TA < TB Cannot be determined 1 2 3 4 5

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**Torque Applied to a Rigid Body**

A rigid body is a system where the particles are strongly bound to each other and maintain their relative orientation during motion The net torque is the sum of all the indivisual torques applied to the system τnet = Στi = ΣriFi sinθi

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**Newton’s Second Law Applied to Rotating Objects**

τnet = Σri miai sin θi = Σri miat at = rα τnet = Σri mi ri α = Σmi ri2 α τnet = Iα

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**Moment of Inertia Moment of Inertia is defined as I = Σmi ri2**

Moment of Inertia resists changes in rotational motion It depends not just on the mass but how far the mass is located from the axis of rotation

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Ex. Find the moment of inertia for the following system of objects connected by light rods of the given length and rotating about the given axis. .75 kg .20 m 42° 1.0 kg .40 m .10 m .25 kg .075 m .50 kg

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A hoop and a disk (made of different materials) have identical masses and radii. Which has a greater moment of inertia? Hoop Disk They are the same Cannot be determined 1 2 3 4 5

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**Moments of Inertia of Extended Objects**

Different shaped objects have different moments of inertia In general, the further the mass is from the axis, the greater the moment of inertia

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Identical torques are applied to a hoop and a disk of the same mass and radius. How do the resulting angular accelerations compare? αhoop > αdisk αhoop = αdisk αhoop < αdisk Cannot be determined 1 2 3 4 5

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Four engines oriented 90° apart fire cw along the tangent of a 1200 kg cylindrical satellite of radius 2.5 m initially at rest. The engines each exert a thrust of 250 N. a) Sketch the situation. b) Draw a free body diagram for the satellite. c) What is the net force on the satellite? d) What is the net torque on the satellite? e) Find the angular acceleration of the satellite. f) If the rockets fire for 10.0 s, what is the angular velocity of the satellite?

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Center of Gravity The weight of an extended object can exert a torque on the object called the gravitational torque τgrav = Σri mig = Mtotrcg g We can calculate the torque as though it is due to all the weight Mtotg acting at a single point rcg called the center of gravity

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Center of Gravity The center of gravity of an object is the balance point. If the object is supported at the center of gravity, the gravitational torque on it is zero. For a uniform object, the center of gravity coincides with the center of the object

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Static Equilibrium An object in static equilibrium has a velocity of 0 and an angular velocity of 0

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**Which of the following is true about an object in static equilibrium**

Which of the following is true about an object in static equilibrium? (TPS) Fnet = 0 τnet = 0 I = 0 1 and 2 1, 2, and 3 1 2 3 4 5

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**Which of the following is true about an object in static equilibrium**

Which of the following is true about an object in static equilibrium? (TPS) Fnet = 0 τnet = 0 I = 0 1 and 2 1, 2, and 3 1 2 3 4 5

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**Conditions for Static Equilibrium**

Translational Equilibrium – Fnet = 0 Rotational Equilibrium – τnet = 0 If an object is in static equilibrium, then its net torque is 0 regardless about which axis the torque is calculated

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**Ex. A beam of negligible mass is supported at its center by a fulcrum**

Ex. A beam of negligible mass is supported at its center by a fulcrum. A .50 kg mass hangs .30 m from one side of the support. And a .40 kg mass hangs from the other side of the beam. a) Sketch the situation. b) Draw a free body diagram for the beam. c) Find the distance form the center of the .40 kg beam so that equilibrium will result.

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**Ex. A uniform board of length 2. 0 m and mass 5**

Ex. A uniform board of length 2.0 m and mass 5.0 kg is supported by a fulcrum at .35 m from one end. a) Sketch the situation. b) On which end must a mass be placed so that equilibrium results. c) How much mass must be placed at that end of the board so that equilibrium results?

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**Rotational Work and Energy**

A spool with a string can be wound up to lift a weight Rotating objects can do work

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**Rotational Kinetic Energy**

In a rotating object each atom instantaneously moves along the tangent Consequently, objects have kinetic energy when they rotate

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**Rotational Kinetic Energy**

The total kinetic energy of a rotating object is KErot = Σ1/2 mivi2 But we can relate the tangential speed to the angular velocity vt = ω r KErot = Σ1/2 miω2ri2 = (Σ1/2 miri2) ω2 KErot =1/2 Iω2

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**Rotational Work Energy Theorem**

Rotational work changes rotational KE τθ = Δ1/2 I ω2

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**FPE - Ex. A hoop of mass 0. 5 kg and radius**

FPE - Ex. A hoop of mass 0.5 kg and radius .25 m is initially rotating at 20.0 s-1. Friction applies a torque of -2.0 Nm bringing the hoop to rest. a) Sketch the situation. b) Find the amount of work done by the friction. c) Through what angular displacement does the hoop turn?

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**Total Kinetic Energy of a Rolling Object**

If an object slides it has KE = ½ mv2 If an object rotates it has KE = ½ Iω2 If an object is rolling it has both translational (sliding) and rotational kinetic energy KErolling = ½ mv2 + ½ Iω2

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A hoop and disk of the same mass and radius are released from rest at the top of a ramp. If both roll without slipping, which will reach the bottom of the ramp first? (TPS) Hoop Disk Tie Cannot be determined 1 2 3 4 5

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A hoop and disk of the same mass and radius are released from rest at the top of a ramp. If both roll without slipping, which will reach the bottom of the ramp first? (TPS) Hoop Disk Tie Cannot be determined 1 2 3 4 5

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A hoop and solid sphere of the same mass and radius are released from rest at the top of a ramp. If both roll without slipping, which will reach the bottom of the ramp first? (TPS) Sphere Hoop Tie Cannot be determined 1 2 3 4 5

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A hoop and solid sphere of the same mass and radius are released from rest at the top of a ramp. If both roll without slipping, which will reach the bottom of the ramp first? (TPS) Sphere Hoop Tie Cannot be determined 1 2 3 4 5

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A solid sphere and disk of the same mass and radius are released from rest at the top of a ramp. If both roll without slipping, which will reach the bottom of the ramp first? (TPS) Sphere Disk Tie Cannot be determined 1 2 3 4 5

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A solid sphere and disk of the same mass and radius are released from rest at the top of a ramp. If both roll without slipping, which will reach the bottom of the ramp first? (TPS) Sphere Disk Tie Cannot be determined 1 2 3 4 5

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Angular Momentum The angular moment of a rigid body rotating about an axis is L = Iω SI Units?

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**Ex. A thin rectangular sheet of mass**

Ex. A thin rectangular sheet of mass .50 kg and width perpendicular to the axis of .20 m rotates about an axis through its center at 15 s-1. a) Sketch the situation. b) Find the angular momentum of the board. c) What would we need to do to change the angular momentum of the board?

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**Angular Momentum of a System of Objects**

If we have a system of objects than Ltotal = ΣLi To change the angular momentum of the system we exert a torque ΔL/Δt = τ τtotal = Σ τi = Σ τi, ext + Σ τi, int But Σ τi, int = 0 so τtotal = Σ τi, ext = τnet, ext

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**Conservation of Angular Momentum**

ΔLtotal/Δt = τnet, ext If τnet, ext = 0 then ΔLtotal/Δt = 0 If the net external torque on a system is 0, then the total angular momentum of the system is conserved

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**Ex. A figure skater is spinning on ice. a) Describe the system**

Ex. A figure skater is spinning on ice. a) Describe the system. b) Are there any external torques acting on the system? c) If she brings her arms in, is she exerting an internal or external torque. d) What happens to her angular velocity when she brings her arms in? e) Explain in terms of conservation of angular momentum.

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**Ex. A disk of mass. 45 kg and radius. 20 m is initially spinning at**

Ex. A disk of mass .45 kg and radius .20 m is initially spinning at .40 s-1 about an axis through the center of the circular face. A hoop of the same mass and radius is dropped from rest concentrically onto the disk so that the two stick together. a) Sketch the situation. b) Is the torque between the hoop and disk an external or internal torque to the system of hoop and disk? c) Is angular momentum conserved? d) Find the final angular velocity of the system.

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