Rotation and Translation Related

As with all dynamics problems, first start with a detailed diagram showing all forces and accelerations involved ( shown at the left).

Summing the forces acting on the spool give

    ΣF_x => f_A = ma_x

    ΣF_y => N_A - mg = 0

Since the spool rotates about its center of gravity, the moments are summed at the center, giving

    ΣM_cg => f_Ar = I_cg α

There are four unknowns (f_A, a_x, N_A, α) but only three equations. Thus, to complete the solution, a fourth equation must be derived that relates the acceleration of the spool's center of gravity to the angular acceleration.

Because the spool does not slip, the acceleration of point A can be related to the acceleration of the cg point,

     a_cg = a_A + a_cg/A

     a_cg-xi + 0j = [a_A-xi + a_A-y j] + [a_cg-xi - a_cg-y j]

     a_xi + 0j = [a_trucki + ω²r j] + [α r i - ω²r j]

Equating i terms give

     a_x = a_truck - rα

There are now four equations to solve for the four unknowns (f_A, a_x, N_A, α). Combining gives two equations,

     (ma_x) r = I_cgα
     a_x = a_truck - rα

Reducing to one equation,

     mr (a_truck - rα) = I_cgα

or

     α = m r a_truck / [I_cg + mr²]
        = 140 (0.5) (1) / [17.5 + 140 (0.5)²]
        = 1.333 rad/s²

and

      a_x = a_truck - rα = 1 - 0.5(1.333) = 0.3333 m/s

Thus, the spool accelerates to in the same direction as the truck acceleration even though it rolls backwards with an angular acceleration of 1.333 rad/s².

Linear Acceleration Direction

To determine the time it takes for the spool to reach the end of the truck, it helps to draw a diagram of the spool at the end of the truck.

From the diagram, when the spool reaches the end of the truck, the following relationship is true,

     x_truck - 3.5 m = x_cg

Use the acceleration of the center of gravity of the spool and the acceleration of the truck to integrate to find x_cg and x_truck,

     x_cg(t) = 0.167 t²

     x_truck(t) = 1/2 t²

Substituting the expressions for x_cg and x_truck, gives

     1/2 t² - 3.5 = 0.167 t²

Solving for t,

     t = 3.24 s