Lab 8

Provided data for Exp 8 and instructions for data analysis and lab report

1. Provided data for Exp 8.1

Measured height of the inclined plane (used in Exp 8.1 video): h = 0.139 m

2. Instruction for data analysis in Exp 8.1

(a) Watch the video of Exp 8.1: Race two round rigid bodies with same c values and with different c

values. Record the results (faster or same) in Table 1 and Table 2 (on P67 in the lab manual).

(b) Calculate the speeds at the bottom of the inclined plane use Eq. (4) and c values given in Fig. 1

(on P65 in the lab manual). Record the calculated speeds in Table 1 (on P67 in the lab manual).

3. Provided data for Exp 8.2

Table 1 Measured masses and dimensions of the 4 objects Middle-step

Pulley

Rotational

Disk

Hollow Sphere Ring

Radius (m) 0.0142 0.0472 0.0375 inner: 0.0270

outer: 0.0382

Mass (kg) -----------

0.1211 0.1354 0.4674

Provided fitting data for Exp 8.2 are given in Fig. 1 in which the #’s correspond to the Measurement #.

(#1)

(#4)

(#2) (#3)

Data of from Measurement #1 to #4

4. Instruction for data analysis in Exp 8.2

(a) Read the purpose of each Measurement from #1 to #4 in the lab manual.

Note: the #s shown in Fig. 1 correspond to the Measurement #s.

(b) From the fitting slopes shown in Fig. 1 to determine the angular accelerations and record the

results in Table 2 (on P72 in the lab manual).

(c) Follow the instruction in the section “Data analysis” on P74 and P75 in the lab manual to

determine all the required quantities and record all your results in Table 3 to Table 7 (on P74 and

P75 in the lab manual).

(d) Attention:

• When you calculate (theory)ringI in Table 5 you should use the formula 2 2

1 2

1 ( )

2 I M R R= + for the

annular cylinder which is different from the thin-walled cylinder as shown in the following figures.

• When you calculate (exp.)ringc in Table 6 you should use the formula

1 2

2 (exp.)

(exp)

2  

ring

ring

I R R c with R

MR =

+ , because the thin-walled cylinder is assumed very thin so

that 1 2 1 2

2

R R R R R

+ → .

5. Instructions for lab report

(a) Tables 1 and 2 in Experiment 8.1 should be included in your lab report.

(b) Tables 1 to 7 in Experiment 8.2 should be included in your lab report.

(c) Provided Fig. 1 in Experiment 8.2 should be included in your lab report.

(d) It is required that the answers or solutions to the 3 questions (at the end of the lab manual)

should be included in your lab report.

(e) The required other contents and format for your lab report can be found in the syllabus

R1

R2

Annular cylinder

Thin-walled

hollow cylinder

R

Translational and rotational motion

Linear Displacement:Δ x⃗ = x⃗ − x⃗0 . Unit: m .

Angular displacement:θ = arc length

radius = s

r . Also d θ =

d s

r .

Sign ofθ : ‘+’ if counter-clockwise, ‘-’ if clockwise.

Unit:1rad = 360 ̊ 2π

= 57.3 ̊ .

Linear Velocity:⟨ϑ⃗ ⟩ = Δ x⃗ Δ t and ⃗ϑ = lim

Δ t→0

Δ x⃗ Δ t

= d x⃗

d t .

Unit: m / s .

Angular velocity:⟨ω⟩ = Δθ Δ t

= θ − θ0 t−t0

and ω = lim Δ t→0

Δθ Δ t

= dθ d t .

Sign ofω : ‘+’ if counter-clockwise, ‘-’ if clockwise. Unit: rad /s .

Translational and rotational motion (contd.)

The direction of the angular velocity can be determined by the right hand thumb rule.

Right hand rule: curl 4 fingers other than thumb in the rotating direction. Your thumb then points in direction of ⃗ω . (⃗ω ||±rotatingaxis )

Translational and rotational motion (contd.)

Linear Acceleration:⟨ a⃗⟩ = Δ ϑ⃗ Δ t and ⃗a = lim

Δ t→0

Δ ϑ⃗ Δ t

= d ϑ⃗

d t

Unit: m / s2 .

Angular acceleration:⟨α⟩ = Δω Δ t

= ω − ω0

t − t 0 and

α = lim Δ t→0

Δω Δ t

= dω d t

= d 2θ d t

2 , ⃗α =

d ω⃗ d t

Sign ofα : ‘+’ if ⃗α || ω⃗ , ‘-’ if ⃗α ||−ω⃗ (⃗α ||±rotatingaxis ). Unit: rad /s2 .

Force:⃗F = ma⃗ . Unit: N . Torque:⃗τ = I α⃗ . Direction from: ⃗τ = r⃗ × F⃗ . If direction in know, rotation is obtained from Right Hand Rule. Unit: N m .

Moment of Inertial

Newton’s second law for rotational motion about a fixed axis is τ⃗ = I α⃗ , (1)

whereI is the moment of inertia, and ⃗τ is net torque. ThisI is the counterpart ofm in rotational motion. For a single particle of massm ,

rotating about an axis in a fixed circular orbit of radius r, moment of inertia is given by:

I=mr2 (2) For extended rigid body, made

of many such particles, moment of inertia is given by

I=∑ i

mi ri 2 (3)

r

Axis of rotation

Body of mass m

Fig. a Single particle rotation

Rigid body rotation

For spherical or cylindrical rigid bodies (Fig. 1), we can write the moment inertia to be:

I=c M R2 (4) whereM is the total mass of the body,R is the radius w.r.t to the axis of rotation and c is a constant that depend on the shape and inner structure of the body, and the axis of rotation chosen. Examples:

Rigid body rolling down inclined plane (8.1)

Key Idea: Two bodies of different shapes, sizes and materials are rolled down an inclined plane, starting from rest at heighth . Observation: Only c values of the bodies determine which

one reaches the bottom first.

Deriving the velocity equation

P. E. :U , Translational K.E. :KT , Rotational K.E. :KR . At the top (body at rest at height h ): We haveU = M gh , KT = 0 , KR = 0 . At the bottom (body in motion at height0 ):

We haveU = 0 , KT = 1

2 M ϑcm

2 , KR = 1

2 Iωcm

2 .

But, we can writeωcm= ϑcm

R and I=c M R2 .

So, Rotational K.E = 1 2 c M R

2 ( ϑcmR ) 2

= 1

2 c M ϑcm

2 .

Conservation of energy between top and bottom:

M gh + 0= 0 + 1

2 Mϑcm

2 + 1

2 c M ϑcm

2 → ϑcm = √ 2 gh1 + c

(5)

Setup for for exp 8.2

Equation for 8.2

For experiment 8.2, we use equation from the the manual:

I = mr2 ( grα−1 ) (6)

Here, the different quantities are as follows:

m → mass hanging from the pulley (4.93 gm or 14.93 gm) r → radius of middle step pulley α → angular acceleration you obtain from the slope of the

plots and put in Table 2 g → acceleration due to gravity

Calculation in tables 3-6

Table 3: Use equation (6) above to compute theI sys and I sys+object . Table 4: Use the values from Table 3 to obtain: Iobject (exp .) = I sys+object−I sys

Table 5: Use equation (4) from above to obtain: Iobject (theory)=c M R

2

where M→ mass of object, R→ radius of object. Theoretical values forc are obtained from Figure 1 of Exp 8.1. Table 6: Use values from Table 4 to compute:

cobject(exp .) = I object (exp .)

M R 2

NOTE: For the Ring: R= (inner radius) + (outer radius)

2 .

End of Theory