Lab 8 | Physics homework help

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 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

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