Lab 5 Torque and Angular Acceleration - Andrews … 5 Torque and Angular Acceleration Objective: <...

Lab 5 Torque and Angular Acceleration

Objective:

< To observe the relationship between torque, angular acceleration and moment of inertia.

Equipment:

< Rotating table< Ring and disk< Hooked weight sets< Dial caliper< Pasco Smart Pulley< Pasco Science Workshop and Computer Interface

Physical principles: Newton’s law of motion for rotation asserts that the net torque acting on an object equals theproduct of its moment of inertia and its angular acceleration.

(1)τ α= I

Torque is defined as the product of the component of force perpendicular to the lever arm and thelength of the lever arm.

(2)τ = ⊥F l

The moment of inertia is defined as the product of each mass piece times the square of that massfrom the axis of rotation.

(3)I m ri= ∑ ⊥2

When an object is rotating the tangential acceleration of a point on its surface is equal to theangular acceleration of the object times the distance of the point from the axis of rotation.

(4)a r= α

The Experiment:

A rotating platform is accelerated by a string that is wrapped on a drum attached to it. The radiusof this drum is denoted by b and is determined as half of the diameter of the drum as measuredwith the dial caliper. The string passes over a Smart Pulley and around a second pulley that supports an accelerating mass m. The string is secured to a Force Sensor that measures the force

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

Objective:

Equipment:

(1)τ α= I

(2)τ = ⊥F l

(3)I m ri= ∑ ⊥2

(4)a r= α

The Experiment:

applied to the drum. Since this force acts tangentially to the drum the torque that it producesabout the drum axis is given by

(5)τ = F b

From the definition of the moment of inertia the moment of inertial of a system of two parts isthe sum of the moments of inertia of the parts. The table platform has a moment of inertia Ip. Inthis experiment a disk or a thick ring are placed on the platform which is then given a set ofaccelerations by hanging different masses on the movable pulley. The Smart Pulley measures thevelocity of the string as it passes over the pulley. This is the same as the tangential velocity ofpoints on the surface of the drum. A graph of this velocity versus time gives the tangentialacceleration of the drum surface which from equation (4) with r = b gives the angularacceleration of the platform.

(6)α = a

A graph of torque computed from equation (5) and angular acceleration computed from equation(6) will be a straight line through the origin with a slope equal to the moment of inertia of therotating system if Newton’s law of motion for rotation is correct. Such a graph for the emptyplatform gives the moment of inertia of the platform, Ip. A similar graph when the platform has adisk or thick ring on it gives the total moment of inertia Itot = Ip + I where I is the moment ofinertia of the disk or thick ring. The values I = Itot - Ip can then be compared with those calculatedfrom the formulae

(7)( )I M R I M R Rdisk thick ring o i= = +1

22 2 2

where R is the radius of the disk, Ro is the outer radius of the thick ring and Ri is the inner radiusof the thick ring. These radii are determined as half the diameter as measured with a ruler. Themass of each is labeled by M and is indicated on the disk and ring.

Procedure:

General Measurements. Using the dial calipers measure and record the drum diameter d. Calculate and record the drum radius. Record the masses of the disk and thick ring. With aruler measure and record the disk diameter, the inner thick ring diameter and the outer thick ringdiameter. Calculate and record the disk diameter, the inner thick ring diameter and the outerthick ring diameter. From equation (7) calculate and record the moments of inertia of the diskand thick ring.

Science Workshop set up. Plug the smart pulley phone plug into digital channel 1 and the forcesensor into analogue channel A. Open Science Workshop. Click and drag the smart pulley icononto digital channel 1 and double click on Smart Pulley. To calibrate the smart pulley double

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

(5)τ = F b

(6)α = a

22 2 2

Procedure:

click on the smart pulley icon and set the pulley radius to .0155, then click on OK. Click anddrag the din connector icon onto analogue channel A and double click on Force Sensor. Tocalibrate the force sensor double click on the force sensor icon. Enter 0 in the Low Value box and4.9 in the High Value box. With no weight on the sensor click on Read in the Low Value row. Hand a 500 gram mass on the force sensor and click on Read in the High Value row. Click onSampling Options and set a stop time of 3 s and click on OK twice. Drag the graph icon onto thesmart pulley icon and select velocity and click on Display. Click on the statistics icon E on thelower left and then on the graph E and select Curve Fit and Linear Fit. Click on the new graphicon on the lower left

Platform Moment of Inertia. With the empty platform secured so that it cannot rotate hang 50 gof mass from the movable pulley. Release the platform and immediately click on the REC icon. Record the accelerating mass of .05 kg, the slope of the graph, a2, and the average force in thefirst three columns of table I. Repeat this for masses of 100 g and 150 gm and complete Table I. Open Graphical Analysis and make a graph of torque, J, versus ". Label the columns and titleyour graph. Click on the regression icon to get a linear fit and record the slope of this graph as Ip.

Platform with Disk. Place the disk on the platform and repeat the steps for the empty platformusing masses of 100 g, 200 g, 300 g, 400 g, and 500 g. Subtract the moment of inertia of theplatform that was determined from the data in table I to obtain a measured value of the momentof inertia of the disk. Compute the percentage error from

percent errorI I

disk slope disk calc

disk calc

= ⋅−

Platform with Thick Ring. Remove the disk and place the thick ring on the platform. Repeat thesteps for the platform with disk.

Discuss your results in your conclusion. Is Newton’s law of motion for rotation consistent withyour observations? Do the slopes of the torque versus angular acceleration graphs predictmoments of inertia that agree with your calculated ones? Discuss sources of error. The presenceof a constant frictional torque would shift your graphs upward. Did your graphs indicate thatfrictional torques were significant?

percent errorI I

disk calc

= ⋅−

percent errorI I

disk calc

= ⋅−

percent errorI I

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= ⋅−

percent errorI I

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= ⋅−

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disk calc

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percent errorI I

disk calc

= ⋅−

d = b = d/2 = Mdisk = Mthick ring =

Ddisk = Rdisk = Ddisk/2 = Idisk =

Douter = Router = Douter/2 =

Dinner = Rinner = Dinner/2 = Ithick ring =

Table I Empty Platform

m a F " = a/b J = F b

Table II Platform with Disk

m a F " = a/b J = F b

Itot =

Table III Platform with Thick Ring

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

m a F " = a/b J = F b

Itot =

Lab 5 Torque and Angular Acceleration - Andrews … 5 Torque and Angular Acceleration Objective: <...

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Transcript of Lab 5 Torque and Angular Acceleration - Andrews … 5 Torque and Angular Acceleration Objective: <...