Consider a cylinder of length L and radius R with density given by p(r) = po"/R: a) What is the mass of the cylinder? b) What is the rotational inertia about the axis of the cylinder? c) Suppose you lay the cylinder down on its side on a pool table. Where do you strike the cylinder with the cue such that it begins rolling smoothly and doesn’t spin?

(a) Let the given cylinder’s volume be denoted by V. Its total mass (M) can be determined by integrating its density (ρ) expression over the cylindrical volume in the cylindrical coordinates as follows: ρ=dMdVM=∫dM=∫ρrdV=∫0Rρ0rRrdr∫02πdϕ∫0Rdz=ρ0RR332πL=23πρ0R2L(b) The given cylinder’s rotational inertia (I) may be determined from its formula as follows: I=∫r2dM=∫r2ρrdV ∵ρ=dMdV=∫0Rr2ρ0rRrdr∫02πdϕ∫0Ldz=ρ0RR552πL=25πρ0R4L(c) Consider the following diagram of the cylinder’s cross-section: Here, the applied force (F) acts horizontally on the given cylinder at the height (h) from the cylinder’s center, which causes linear accelerations a0 and a at the cylinder’s center and at the distance (h) from the center. The radius (R) from the cylinder’s center to the point, where F acts, makes the angle (θ) with the horizontal.If F is applied at any point on the cylinder’s lower half, the linear acceleration caused due to the cylinder’s rotation would oppose F and the cylinder will likely slip. For the cylinder not to slip and just roll, the torque (τ) due to F must cause the linear accelerations a0 and a at the cylinder’s center and at the distance h above the center, respectively, so that they can be related by the relationship between the linear and the angular acceleration (α). The slip condition must be so that a and a0 are equal. Find the condition for h from this as follows: τ=RFsinθ=hF ∵sinθ=hRa=a0hτI=FM ∵a=hα, τ=Iα, F=Ma0hhF25πρ0R4L=F23πρ0R2L ∵I=25πρ0R4L, M=23πρ0R2Lh=35R