The truck shown below is initially at rest with a solid cylindrical roll of paper sitting on its bed. If the truck moves forward with a uniform acceleration a, what distance s does it move before the paper rolls off its back end? (Hint: If the roll accelerates forward with a', then it accelerates backward relative to the truck with an acceleration a − a'. Also, R? = a − a'. Use the following as necessary: d.) The image depicts a red truck with a flatbed carrying a large cylindrical object. The flatbed length is marked with a horizontal line labeled "d," indicating the distance from the front of the flatbed to the object. An arrow labeled "a" points to the right, above the truck, indicating the direction of the truck's acceleration. The driver is visible in the cabin. This diagram illustrates concepts of dynamics, such as acceleration and motion, commonly studied in physics.

Consider the acceleration of the roll with respect to the ground is a and its acceleration with respect to the truck it has the acceleration is a-a'. Write the relation for the angular acceleration of the truck. Rα=a-a'α=a-a'R 1 The moment of inertia of the cylindrical paper roll is, I=12MR2 2 The force of friction makes the paper roll to rotate backward. Hence, the net force on the cylindrical paper roll is given by ∑F=fs=ma' 3 Write the relation for the torque on the paper roll. τ=Rfs Rfs=Iα=Ia-a'R=12MR2a-a'Rfs=Ma-a'2 4 Solve the equations (3) and (4) for the acceleration a'. Ma'=Ma-a'2a'=a-a'22a'=a-a'3a'=aa'=a3 5 Calculate how long the paper roll takes off the back end of the truck. d=v'0t+12a-a't2d=0+12a-a/3t2=at23t=3da 6 Calculate the distance that the truck will travel during the interval of t. x-x0=v0t+12at2s=0+12a3da2=3d2=1.5d Therefore, the distance s the truck move before the paper rolls off its back end is 1.5d.