AP Physics 1: 7.1 Rotational Kinematics – Exam Style questions with Answer- FRQ

Question

A billiard ball has mass M, radius R, and moment of inertia about the center of mass Ic = 2 MR²/5 The ball is struck by a cue stick along a horizontal line through the ball’s center of mass so that the ball initially
slides with a velocity vo as shown above. As the ball moves across the rough billiard table (coefficient of sliding friction μk), its motion gradually changes from pure translation through rolling with slipping to rolling without slipping.
a. Develop an expression for the linear velocity v of the center of the ball as a function of time while it is rolling with slipping.
b. Develop an expression for the angular velocity ω of the ball as a function of time while it is rolling with slipping.
c. Determine the time at which the ball begins to roll without slipping.
d. When the ball is struck it acquires an angular momentum about the fixed point P on the surface of the table.
During the subsequent motion the angular momentum about point P remains constant despite the frictional force. Explain why this is so.

▶️Answer/Explanation

Ans:

a. \(\sum F\)= ma; Ff = μFN;
– μMg = Ma
a = –μg
v = v0 + at
v = v0 – μgt
b. τ = Iα where the torque is provided by friction Ff = μMg
    μMgR = (2MR2/5)α
α = (5μg/2R)
ω = ω0 + αt = (5μg/2R)t
c. Slipping stops when the tangential velocity si equal to the velocity of the center of mass, or the condition for
    pure rolling has been met: v(t) = ω(t)R
    v0 – μgt = R(5g/2R)t, which gives T = (2/7)(v0/μg)
d. Since the line of action of the frictional force passes through P, the net torque about point P is zero. Thus, the time rate of change of the angular momentum is zero and the angular momentum is constant.

Question

A system consists of two small disks, of masses m and 2m, attached to a rod of negligible mass of length 3l as shown above. The rod is free to turn about a vertical axis through point P. The two disks rest on a rough horizontal surface; the coefficient of friction between the disks and the surface is μ. At time t = 0, the rod has an initial counterclockwise angular velocity ωo about P. The system is gradually brought to rest by friction. Develop expressions for the following quantities in terms of μ m, l, g, and ωo a. The initial angular momentum of the system about the axis through P
b. The frictional torque acting on the system about the axis through P
c. The time T at which the system will come to rest.

▶️Answer/Explanation

Ans:

a. L = Iω where I = \(\sum \)mr2 = (2m)l2 + m(2l)2 = 6ml2
    L = 6ml2ω
b. Ff = μmg
\(\sum \)τ = –(μ(2m)gl + μmg(2l)) = –4μmgl
c. α = τ/I = –4μmgl/6ml2 = –2μg/3l
  ω = ω0 + αt; setting ω = 0 and solving for T gives T = 3ω0l/2μg

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