Engineering Dynamics · Impulse & Momentum · Open Problem Companion

Chapter Nineteen
Planar Kinetics of a Rigid Body: Impulse and Momentum

19.1 Fundamentals 12 problems
Concepts
  1. 1.Determine the angular momentum of the \(100\,\mathrm{kg}\) disk or rod about point \(G\) and about point \(O\). figure — add your own
  2. 2.Determine the angular impulse about point \(O\) for \(t = 3\,\mathrm{s}\). figure — add your own
  3. 3.The \(60\,\mathrm{kg}\) wheel has radius of gyration about its center \(O\) of \(k_O = 300\,\mathrm{mm}\). If subjected to a couple moment \(M = (3t^2)\,\mathrm{N\cdot m}\), determine angular velocity when \(t = 4\,\mathrm{s}\), starting from rest. figure — add your own
  4. 4.The \(300\,\mathrm{kg}\) wheel has \(k_O = 400\,\mathrm{mm}\). Subjected to \(M = 300\,\mathrm{N\cdot m}\), determine angular velocity after \(6\,\mathrm{s}\) from rest (no slipping) and the friction force from the ground. figure — add your own
  5. 5.Rod \(OA\) (negligible mass) is subjected to \(M = 9\,\mathrm{N\cdot m}\). Determine angular velocity of the \(10\,\mathrm{kg}\) inner gear after \(5\,\mathrm{s}\) from rest. The gear has \(k_A = 100\,\mathrm{mm}\) and rolls on the fixed outer gear \(B\) in the horizontal plane. figure — add your own
  6. 6.Gears \(A\) and \(B\) have masses \(10\,\mathrm{kg}\) and \(50\,\mathrm{kg}\) with radii of gyration \(k_A = 80\,\mathrm{mm}\), \(k_B = 150\,\mathrm{mm}\). Gear \(A\) is subjected to \(M = 10\,\mathrm{N\cdot m}\) from rest. Determine angular velocity of gear \(B\) when \(t = 5\,\mathrm{s}\). figure — add your own
  7. 7.The \(50\,\mathrm{kg}\) spool is subjected to horizontal force \(P = 150\,\mathrm{N}\). If it rolls without slipping, determine its angular velocity \(3\,\mathrm{s}\) after starting from rest. \(k_G = 175\,\mathrm{mm}\). figure — add your own
  8. 8.The reel has mass \(150\,\mathrm{kg}\), \(k_G = 1.25\,\mathrm{m}\), and is subjected to torque \(M = 250\,\mathrm{N\cdot m}\). Determine angular velocity after 3 s from rest. Coefficient of kinetic friction with the horizontal plane is \(\mu_k = 0.15\). figure — add your own
  9. 9.Show that for a rigid body of mass \(m\) rotating about a fixed point \(O\), the momenta of all particles can be represented by a single vector \(m\mathbf{v}_G\) acting through the center of percussion \(P\), where \(r_{P/G} = k_G^2 / r_{G/O}\).
  10. 10.Show that for a body with linear momentum \(\mathbf{L} = m\mathbf{v}_G\) and angular momentum \(\mathbf{H}_G = I_G\boldsymbol{\omega}\), the angular momentum about the instantaneous center \(IC\) is \(\mathbf{H}_{IC} = I_{IC}\boldsymbol{\omega}\).
  11. 11.Show that if a slab rotates about a fixed axis through its mass center \(G\), the angular momentum is the same when computed about any other point \(P\).
  12. 12.The \(40\,\mathrm{kg}\) disk is rotating at \(\omega = 100\,\mathrm{rad/s}\) when force \(P\) is applied to the brake as shown by the graph. If \(\mu_k = 0.3\) at \(B\), determine the time needed to stop the disk. Neglect brake thickness. figure — add your own
19.2 Impulse & Momentum 28 problems
Skills
  1. 13.The \(40\,\mathrm{kg}\) roll of paper rests along the wall with \(\mu_k = 0.2\). If a vertical force \(P = 40\,\mathrm{N}\) is applied, determine angular velocity when \(t = 6\,\mathrm{s}\) from rest. Neglect mass of unraveled paper; \(k_O = 80\,\mathrm{mm}\). figure — add your own
  2. 14.The \(30\,\mathrm{kg}\) gear \(A\) has \(k_O = 125\,\mathrm{mm}\). The \(20\,\mathrm{kg}\) gear rack \(B\) is subjected to \(P = 200\,\mathrm{N}\). Determine time required for the gear to reach \(20\,\mathrm{rad/s}\) from rest. Contact surface is smooth. figure — add your own
  3. 15.A wire of negligible mass is wrapped around the outer surface of the \(2\,\mathrm{kg}\) disk. If released from rest, determine its angular velocity in 3 s. figure — add your own
  4. 16.The airplane travels straight at \(300\,\mathrm{km/h}\) when engines \(A\) and \(B\) produce thrusts \(T_A = 40\,\mathrm{kN}\), \(T_B = 20\,\mathrm{kN}\). Determine angular velocity after \(t = 5\,\mathrm{s}\). Mass \(= 200\,\mathrm{Mg}\), \(k_G = 15\,\mathrm{m}\). figure — add your own
  5. 17.Determine the height \(h\) of the pool table bumper so that when the ball of mass \(m\) strikes it, no frictional force develops at \(A\). Assume the bumper exerts only a horizontal force. figure — add your own
  6. 18.The impact wrench consists of a slender 1‑kg rod AB (580 mm long) and cylindrical end weights at A and B, each 1 kg. The assembly is free to turn. If rod AB is given \(\omega = 4\,\mathrm{rad/s}\) and it strikes bracket C without rebounding, determine the angular impulse imparted to the lug nut. figure — add your own
  7. 19.If the ball has weight \(W\), radius \(r\), and is thrown onto a rough surface with velocity \(\mathbf{v}_0\) parallel to the surface, determine the backspin \(\omega_0\) needed so that it stops spinning at the same instant its forward velocity is zero. (Coefficient of friction not needed.) figure — add your own
  8. 20.The double pulley consists of two wheels attached together. Mass \(15\,\mathrm{kg}\), \(k_O = 110\,\mathrm{mm}\). Block \(A\) has mass \(40\,\mathrm{kg}\). Determine speed of block in 3 s after a constant force of \(2\,\mathrm{kN}\) is applied to the rope wrapped around the inner hub. Block originally at rest. figure — add your own
  9. 21.The square plate has mass \(m\) and is suspended at corner \(A\) by a cord. If it receives a horizontal impulse I at corner \(B\), determine the location \(y\) of point \(P\) about which the plate appears to rotate during impact. figure — add your own
  10. 22.The slender rod has mass \(m\) and is suspended at \(A\) by a cord. If it receives a horizontal blow giving impulse I at its bottom \(B\), determine location \(y\) of point \(P\) about which the rod appears to rotate. figure — add your own
  11. 23.The \(100\,\mathrm{kg}\) reel has \(k_G = 200\,\mathrm{mm}\). If cable \(B\) is subjected to \(P = 300\,\mathrm{N}\), determine time required for the reel to reach \(20\,\mathrm{rad/s}\). \(\mu_k = 0.15\) between reel and plane. figure — add your own
  12. 24.The \(4\,\mathrm{kg}\) slender rod rests on a smooth floor. If kicked so as to receive a horizontal impulse \(I = 8\,\mathrm{N\cdot s}\) at point \(A\), determine its angular velocity and speed of its mass center. figure — add your own
  13. 25.The \(100\,\mathrm{kg}\) spool rests on an inclined surface with \(\mu_k = 0.1\). Determine angular velocity when \(t = 4\,\mathrm{s}\) after release from rest. \(k_G = 0.25\,\mathrm{m}\). figure — add your own
  14. 26.The double pulley (mass \(15\,\mathrm{kg}\), \(k_O = 110\,\mathrm{mm}\)) has block \(A\) (40 kg). Determine speed of block in 3 s after a constant force \(F = 2\,\mathrm{kN}\) is applied to the rope. Block originally at rest. figure — add your own
  15. 27.A \(4\,\mathrm{kg}\) disk \(A\) is mounted on arm \(BC\) (negligible mass). A torque \(M = (5e^{0.5t})\,\mathrm{N\cdot m}\) is applied at \(C\). Determine angular velocity of \(BC\) in 2 s from rest for three cases: (a) disk in smooth bearing (curvilinear translation), (b) disk fixed to shaft, (c) disk given initial spin \(\omega_D = -80\mathbf{k}\,\mathrm{rad/s}\). figure — add your own
  16. 28.The rod of length \(L\) and mass \(m\) lies on a smooth horizontal surface and is subjected to force \(\mathbf{P}\) at its end \(A\). Determine location \(d\) of the point about which the rod begins to turn (zero velocity). figure — add your own
  17. 29.The drum has mass \(70\,\mathrm{kg}\), radius \(300\,\mathrm{mm}\), \(k_O = 125\,\mathrm{mm}\). Coefficients: \(\mu_s = 0.4\), \(\mu_k = 0.3\). Determine angular velocity 2 s after release from rest. \(\theta = 30^\circ\). figure — add your own
  18. 30.A motor transmits torque \(M = 0.05\,\mathrm{N\cdot m}\) to the center of gear \(A\). Determine angular velocity of each of the three equal smaller gears in 2 s from rest. The smaller gears are pinned at their centers; masses and radii of gyration are given. figure — add your own
  19. 31.The crate has mass \(m_c\). Determine the constant speed \(v_0\) it acquires as it moves down the conveyor. The rollers each have radius \(r\), mass \(m\), and are spaced \(d\) apart. Friction causes each roller to rotate when the crate contacts it. figure — add your own
  20. 32.If the shaft is subjected to torque \(M = (15t^2)\,\mathrm{N\cdot m}\), determine angular velocity of the assembly when \(t = 3\,\mathrm{s}\), starting from rest. Rods \(AB\) and \(BC\) each have mass \(9\,\mathrm{kg}\). figure — add your own
  21. 33.The \(30\,\mathrm{kg}\) reel is mounted on the \(20\,\mathrm{kg}\) cart. If the cable wrapped around the inner hub is subjected to \(P = 50\,\mathrm{N}\), determine velocity of the cart and angular velocity of the reel when \(t = 4\,\mathrm{s}\). \(k_O = 250\,\mathrm{mm}\); neglect size of small wheels. figure — add your own
  22. 34.The double pulley (mass \(15\,\mathrm{kg}\), \(k_O = 110\,\mathrm{mm}\)) has block \(A\) (40 kg) and container \(B\) (85 kg including contents). Determine speed of the container when \(t = 3\,\mathrm{s}\) after release from rest. figure — add your own
  23. 35.The \(30\,\mathrm{kg}\) gear is subjected to \(P = (20t)\,\mathrm{N}\). Determine angular velocity at \(t = 4\,\mathrm{s}\), starting from rest. Gear rack \(B\) is fixed; \(k_O = 125\,\mathrm{mm}\). figure — add your own
  24. 36.The hoop (thin ring) has mass \(5\,\mathrm{kg}\) and is released down the inclined plane with backspin \(\omega = 8\,\mathrm{rad/s}\) and center velocity \(v_G = 3\,\mathrm{m/s}\). \(\mu_k = 0.6\). Determine how long the hoop rolls before it stops slipping. figure — add your own
  25. 37.A man has moment of inertia \(I_z\) about the \(z\) axis. He is at rest on a freely turning platform. If handed a wheel rotating at \(\omega\) with moment of inertia \(I\), determine his angular velocity if he holds it (a) upright, (b) turned out \(\theta = 90^\circ\), (c) turned downward \(\theta = 180^\circ\). Neglect distance \(d\). figure — add your own
  26. 38.The satellite has mass \(200\,\mathrm{kg}\), \(k_z = 0.1\,\mathrm{m}\) (excluding panels). Each solar panel has mass \(15\,\mathrm{kg}\) and is a thin plate. If originally spinning at \(\omega_z = 0.5\,\mathrm{rad/s}\) when \(\theta = 90^\circ\), determine rate of spin when both panels are raised to \(\theta = 0^\circ\). figure — add your own
  27. 39.The \(75\,\mathrm{kg}\) gymnast lets go of the bar in a fully stretched position \(A\) rotating with \(\omega_A = 3\,\mathrm{rad/s}\). Estimate angular velocity when he assumes a tucked position \(B\) (model as uniform slender rod and uniform circular disk respectively). figure — add your own
  28. 40.The \(2\,\mathrm{kg}\) rod \(ACB\) supports two \(4\,\mathrm{kg}\) disks at its ends. Both disks are given a clockwise angular velocity of \(5\,\mathrm{rad/s}\) while the rod is held stationary, then released. Determine angular velocity of the rod after both disks have stopped spinning relative to the rod due to friction at pins \(A\) and \(B\). Motion in horizontal plane; neglect friction at \(C\). figure — add your own
19.3 Impact & Angular Momentum 25 problems
Skills Plus
  1. 41.The \(80\,\mathrm{kg}\) man holds two dumbbells while standing on a turntable of negligible mass. When arms are extended, turntable rotates at \(0.5\,\mathrm{rev/s}\). Determine angular velocity when he retracts his arms. Model arms as uniform \(6\,\mathrm{kg}\) rods (650 mm long), body as \(68\,\mathrm{kg}\) solid cylinder (400 mm diameter), retracted as \(80\,\mathrm{kg}\) cylinder (450 mm diameter). Dumbbells: two \(5\,\mathrm{kg}\) spheres. figure — add your own
  2. 42.The turntable \(T\) of a record player has mass \(0.75\,\mathrm{kg}\), \(k_z = 125\,\mathrm{mm}\). It turns freely at \(\omega_T = 2\,\mathrm{rad/s}\) when a \(50\,\mathrm{g}\) record (thin disk) falls on it. Determine final angular velocity just after the record stops slipping. figure — add your own
  3. 43.A \(10\,\mathrm{g}\) bullet with velocity \(800\,\mathrm{m/s}\) is fired into the edge of the \(5\,\mathrm{kg}\) disk. Determine angular velocity of disk just after the bullet becomes embedded. Also calculate angle \(\theta\) the disk swings when it stops. Disk initially at rest; neglect mass of rod \(AB\). figure — add your own
  4. 44.Repeat the previous problem, but now the rod \(AB\) has mass \(3\,\mathrm{kg}\). figure — add your own
  5. 45.The circular disk has mass \(m\) and is suspended at \(A\) by a wire. If it receives a horizontal impulse I at edge \(B\), determine the location \(y\) of point \(P\) about which the disk appears to rotate. figure — add your own
  6. 46.A \(7\,\mathrm{g}\) bullet with velocity \(800\,\mathrm{m/s}\) is fired into the edge of the \(5\,\mathrm{kg}\) disk. Determine angular velocity just after embedding, and how far \(\theta\) the disk swings until it stops. Disk initially at rest. figure — add your own
  7. 47.The vertical shaft rotates with angular velocity \(3\,\mathrm{rad/s}\) when \(\theta = 0^\circ\). If force \(\mathbf{F}\) is applied to the collar so that \(\theta = 90^\circ\), determine angular velocity of the shaft and work done by \(\mathbf{F}\). Neglect masses of rods \(GH\), \(EF\), and collars \(I,J\). Rods \(AB\) and \(CD\) each have mass \(10\,\mathrm{kg}\). figure — add your own
  8. 48.The slender bar of mass \(m\) pivots at support \(A\) when released from rest in the vertical position. When it falls and rotates \(90^\circ\), pin \(C\) strikes support \(B\), and pin at \(A\) leaves its support. Determine angular velocity immediately after impact. Pin \(B\) will not rebound. figure — add your own
  9. 49.A thin rod of mass \(m\) has angular velocity \(\omega_0\) while rotating on a smooth surface. Determine its new angular velocity just after its end strikes and hooks onto the peg and the rod starts to rotate about \(P\) without rebounding. Solve (a) with parameters, (b) for \(m = 2\,\mathrm{kg}\), \(\omega_0 = 4\,\mathrm{rad/s}\), \(l = 1.5\,\mathrm{m}\). figure — add your own
  10. 50.Determine height \(h\) at which a billiard ball of mass \(m\) must be struck so that no frictional force develops between it and the table at \(A\). Assume the cue \(C\) only exerts a horizontal force \(\mathbf{P}\). figure — add your own
  11. 51.The pendulum consists of a slender \(2\,\mathrm{kg}\) rod AB and \(5\,\mathrm{kg}\) disk. Released from rest without rotating. When it falls \(0.3\,\mathrm{m}\), end \(A\) strikes hook \(S\) providing a permanent connection. Determine angular velocity of pendulum after it has rotated \(90^\circ\). Treat weight during impact as nonimpulsive. figure — add your own
  12. 52.The target is a thin \(5\,\mathrm{kg}\) circular disk that can rotate freely about the \(z\) axis. A \(25\,\mathrm{g}\) bullet traveling at \(600\,\mathrm{m/s}\) strikes the target at \(A\) and becomes embedded. Determine angular velocity of the target after impact. Initially at rest. figure — add your own
  13. 53.The Hubble Space Telescope is powered by two solar panels. Body mass \(11\,\mathrm{Mg}\), \(k_x = 1.64\,\mathrm{m}\), \(k_y = 3.85\,\mathrm{m}\); each panel \(54\,\mathrm{kg}\) thin plate. Due to an internal drive, panels are given angular velocity \(0.6\,\mathrm{rad/s}\) relative to the telescope. Determine angular velocity of the telescope due to panel rotation. Originally traveling at \(\mathbf{v}_G = [-400\mathbf{i} + 250\mathbf{j} + 175\mathbf{k}]\,\mathrm{m/s}\); neglect orbital rotation. figure — add your own
  14. 54.The \(12\,\mathrm{kg}\) rod AB is pinned to the \(40\,\mathrm{kg}\) disk. If the disk is given \(\omega_D = 100\,\mathrm{rad/s}\) while the rod is held stationary, and the assembly is then released, determine angular velocity of the rod after the disk has stopped spinning relative to the rod due to friction at bearing \(B\). Motion in horizontal plane; neglect friction at pin \(A\). figure — add your own
  15. 55.The rod of mass \(m\) and length \(L\) is released from rest without rotating. When it falls a distance \(L\), end \(A\) strikes hook \(S\) providing a permanent connection. Determine angular velocity \(\omega\) after it has rotated \(90^\circ\). Treat weight during impact as nonimpulsive. figure — add your own
  16. 56.The solid ball of mass \(m\) is dropped with velocity \(\mathbf{v}_1\) onto the edge of a rough step. If it rebounds horizontally off the step with velocity \(\mathbf{v}_2\), determine angle \(\theta\) at which contact occurs. Assume no slipping when striking the step. Coefficient of restitution is \(e\). figure — add your own
  17. 57.The wheel has mass \(50\,\mathrm{kg}\), \(k_G = 125\,\mathrm{mm}\). Determine minimum angular velocity \(\omega_1\) so that it strikes the step at \(A\) without rebounding and then rolls over it without slipping. figure — add your own
  18. 58.The wheel (mass \(50\,\mathrm{kg}\), \(k_G = 125\,\mathrm{mm}\)) rolls without slipping with \(\omega_1 = 5\,\mathrm{rad/s}\) before it strikes the step at \(A\). Determine its angular velocity after it rolls over the step. It does not lose contact with the step. figure — add your own
  19. 59.A \(2\,\mathrm{kg}\) mass of putty \(D\) strikes the uniform \(10\,\mathrm{kg}\) plank ABC with velocity \(10\,\mathrm{m/s}\). If the putty remains attached, determine maximum angle \(\theta\) of swing before the plank momentarily stops. Neglect size of putty. figure — add your own
  20. 60.The hammer consists of a \(10\,\mathrm{kg}\) solid cylinder \(C\) and \(6\,\mathrm{kg}\) uniform slender rod \(AB\). Released from rest when \(\theta = 90^\circ\), strikes the \(30\,\mathrm{kg}\) block \(D\) when \(\theta = 0^\circ\). Determine velocity of block \(D\) and angular velocity of hammer immediately after impact. Coefficient of restitution \(e = 0.6\). figure — add your own
  21. 61.The \(20\,\mathrm{kg}\) disk strikes the step without rebounding. Determine the largest angular velocity \(\omega_1\) the disk can have and not lose contact with the step \(A\). figure — add your own
  22. 62.The pendulum consists of a \(15\,\mathrm{kg}\) solid ball and \(6\,\mathrm{kg}\) rod. Released from rest when \(\theta_1 = 90^\circ\). Determine angle \(\theta_2\) after the ball strikes the wall, rebounds, and the pendulum swings up to momentary rest. \(e = 0.6\). figure — add your own
  23. 63.Two children \(A\) and \(B\), each mass \(30\,\mathrm{kg}\), sit at the edge of a merry-go-round rotating at \(\omega = 2\,\mathrm{rad/s}\). Merry-go-round mass \(180\,\mathrm{kg}\), \(k_z = 0.6\,\mathrm{m}\). Determine angular velocity if \(A\) jumps off horizontally in the \(-n\) direction with speed \(2\,\mathrm{m/s}\) relative to the merry-go-round. Then determine angular velocity if \(B\) jumps off horizontally in the \(+t\) direction with speed \(2\,\mathrm{m/s}\) relative. Neglect friction. figure — add your own
  24. 64.The \(20\,\mathrm{kg}\) cylinder \(A\) is free to slide along rod \(BC\). When at \(x = 0\), the \(50\,\mathrm{kg}\) circular disk \(D\) rotates at \(5\,\mathrm{rad/s}\). If the cylinder is given a slight push, determine angular velocity of the disk when the cylinder strikes \(B\) at \(x = 600\,\mathrm{mm}\). Neglect mass of brackets and smooth rod. figure — add your own
  25. 65.A solid ball with mass \(m\) is thrown on the ground such that at impact it has angular velocity \(\omega_1\) and velocity components \((\mathbf{v}_G)_{x1}\), \((\mathbf{v}_G)_{y1}\). If the ground is rough so no slipping occurs, determine components of velocity of its mass center just after impact. Coefficient of restitution is \(e\). figure — add your own
19.4 Conservation & Rigid-Body Systems 16 problems
Applications
  1. 66.A ball of mass \(8\,\mathrm{kg}\) and initial speed \(v_1 = 0.2\,\mathrm{m/s}\) rolls over a 30‑mm‑long depression. Assuming it rolls off the edges of contact first \(A\), then \(B\) without slipping, determine its final velocity \(\mathbf{v}_2\) when it reaches the other side. figure — add your own
  2. 67.The cable is subjected to a force \(P = (50t^2)\,\mathrm{N}\). Determine angular velocity of the spool 3 s after \(P\) is applied, starting from rest. Spool mass \(75\,\mathrm{kg}\), \(k_O = 0.375\,\mathrm{m}\). figure — add your own
  3. 68.The space capsule has mass \(1200\,\mathrm{kg}\), \(I_G = 900\,\mathrm{kg\cdot m^2}\). It travels forward with \(v_G = 800\,\mathrm{m/s}\) and executes a turn using two jets providing constant thrust \(400\,\mathrm{N}\) for \(0.3\,\mathrm{s}\). Determine capsule's angular velocity just after jets are turned off. figure — add your own
  4. 69.The tire has mass \(9\,\mathrm{kg}\), \(k_O = 225\,\mathrm{mm}\). Released from rest and rolls down the plane without slipping. Determine speed of its center \(O\) when \(t = 3\,\mathrm{s}\). figure — add your own
  5. 70.The wheel has mass \(100\,\mathrm{kg}\), \(k_z = 300\,\mathrm{mm}\), rests on smooth horizontal plane. If the belt is subjected to \(P = 200\,\mathrm{N}\), determine angular velocity of the wheel and speed of its center of mass \(O\) three seconds after the force is applied. figure — add your own
  6. 71.The spool has mass \(15\,\mathrm{kg}\), \(k_O = 0.2\,\mathrm{m}\). If a force of \(200\,\mathrm{N}\) is applied to the cord at \(A\), determine angular velocity of the spool in \(t = 3\,\mathrm{s}\) starting from rest. Neglect mass of pulley and cord. figure — add your own
  7. 72.Spool \(B\) is at rest and spool \(A\) is rotating at \(6\,\mathrm{rad/s}\) when the slack in the cord connecting them is taken up. If the cord does not stretch, determine angular velocity of each spool immediately after the cord is jerked tight. \(m_A = 15\,\mathrm{kg}\), \(k_A = 0.25\,\mathrm{m}\); \(m_B = 7.5\,\mathrm{kg}\), \(k_B = 0.18\,\mathrm{m}\). figure — add your own
  8. 73.A thin disk of mass \(m\) has angular velocity \(\omega_1\) while rotating on a smooth surface. Determine its new angular velocity just after the hook at its edge strikes the peg \(P\) and the disk starts to rotate about \(P\) without rebounding. figure — add your own
  9. 74.The space satellite has mass \(125\,\mathrm{kg}\), \(I_c = 0.940\,\mathrm{kg\cdot m^2}\) (excluding four solar panels). Each panel has mass \(20\,\mathrm{kg}\) and is a thin plate. If originally spinning at \(\omega_z = 0.5\,\mathrm{rad/s}\) when \(\theta = 90^\circ\), determine rate of spin if all panels are raised to \(\theta = 0^\circ\) at the same instant. figure — add your own
  10. 75.A bullet of mass \(m\) is fired with velocity \(v\) into the edge of a disk of mass \(M\) and radius \(R\) that is free to rotate about its center. Determine angular velocity just after impact. figure — add your own
  11. 76.A homogeneous rod of mass \(m\) and length \(L\) is pivoted at one end. It is struck at the other end by a horizontal impulse \(J\). Determine angular velocity immediately after impact. figure — add your own
  12. 77.A disk of mass \(m\) and radius \(R\) rolls without slipping on a horizontal surface. If it is given a horizontal impulse \(P\) at its center, determine its angular velocity immediately after. figure — add your own
  13. 78.A uniform sphere of mass \(m\) and radius \(r\) rolls without slipping on a horizontal plane. It is struck by a horizontal impulse at a height \(h\) above the center. Determine the angular velocity and velocity of the center just after impact. figure — add your own
  14. 79.A rectangular plate of mass \(m\) and sides \(a\) and \(b\) is pivoted at one corner. It receives an impulse at the opposite corner. Determine angular velocity immediately after. figure — add your own
  15. 80.A thin hoop of mass \(m\) and radius \(R\) is rolling without slipping on a horizontal surface. A horizontal impulse is applied at the top point. Determine the new angular velocity. figure — add your own
  16. 81.A uniform rod of mass \(m\) and length \(L\) is suspended by a frictionless pivot at one end. It is struck at a point \(d\) below the pivot by a horizontal impulse \(J\). Determine the angular velocity immediately after. figure — add your own
19.5 Review Problems 9 problems
Review
  1. 82.The cable is subjected to \(P = (50t^2)\,\mathrm{N}\). Determine angular velocity of the spool 3 s after \(P\) is applied, starting from rest. Spool mass \(75\,\mathrm{kg}\), radius of gyration \(0.375\,\mathrm{m}\) about center \(O\). figure — add your own
  2. 83.The space capsule has mass \(1200\,\mathrm{kg}\), \(I_G = 900\,\mathrm{kg\cdot m^2}\). It travels at \(v_G = 800\,\mathrm{m/s}\) and executes a turn using two jets providing constant thrust \(400\,\mathrm{N}\) for \(0.3\,\mathrm{s}\). Determine the capsule's angular velocity just after the jets are turned off. figure — add your own
  3. 84.The tire has mass \(9\,\mathrm{kg}\), \(k_O = 225\,\mathrm{mm}\). Released from rest and rolls down the plane without slipping. Determine speed of its center \(O\) when \(t = 3\,\mathrm{s}\). figure — add your own
  4. 85.The wheel has mass \(100\,\mathrm{kg}\), \(k_z = 300\,\mathrm{mm}\), rests on smooth horizontal plane. If the belt is subjected to \(P = 200\,\mathrm{N}\), determine angular velocity of the wheel and speed of its center of mass \(O\) three seconds after force is applied. figure — add your own
  5. 86.The spool has mass \(15\,\mathrm{kg}\), \(k_O = 0.2\,\mathrm{m}\). If force \(200\,\mathrm{N}\) is applied to the cord at \(A\), determine angular velocity of the spool in \(t = 3\,\mathrm{s}\) starting from rest. Neglect mass of pulley and cord. figure — add your own
  6. 87.Spool \(B\) is at rest and spool \(A\) is rotating at \(6\,\mathrm{rad/s}\) when the slack in the cord connecting them is taken up. If the cord does not stretch, determine angular velocity of each spool immediately after the cord is jerked tight. \(m_A = 15\,\mathrm{kg}\), \(k_A = 0.25\,\mathrm{m}\); \(m_B = 7.5\,\mathrm{kg}\), \(k_B = 0.18\,\mathrm{m}\). figure — add your own
  7. 88.A thin disk of mass \(m\) has angular velocity \(\omega_1\) while rotating on a smooth surface. Determine its new angular velocity just after the hook at its edge strikes the peg \(P\) and the disk starts to rotate about \(P\) without rebounding. figure — add your own
  8. 89.The space satellite has mass \(125\,\mathrm{kg}\), \(I_c = 0.940\,\mathrm{kg\cdot m^2}\) (excluding four solar panels). Each panel has mass \(20\,\mathrm{kg}\) and is a thin plate. If originally spinning at \(\omega_z = 0.5\,\mathrm{rad/s}\) when \(\theta = 90^\circ\), determine rate of spin if all panels are raised to \(\theta = 0^\circ\) at the same instant. figure — add your own
  9. 90.A homogeneous sphere of mass \(m\) and radius \(r\) rolls without slipping on a horizontal plane. It is struck by a horizontal impulse at a height \(h\) above the center. Determine the angular velocity and velocity of the center just after impact. figure — add your own