update-book

books/university-physics-B/Module-B9/.ipynb_checkpoints/problemsB9-checkpoint.ipynb

@@ -409,6 +409,357 @@
     "print(sol)"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<hr style=\"height:2px;border-width:0;color:gray;background-color:gray\">"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<h2>Problem B9.6</h2>\n",
+    "    </header>\n",
+    "\n",
+    "What hanging mass must be placed on the cord to keep the pulley from rotating (see the following figure)? The mass on the frictionless plane is 5.0 kg. The mass on the plane is connected to a cord that wraps around the pulley’s inner radius of 20.0 cm. The hanging mass is connected to a cord that wraps around the pulley’s outer radius of 30.0 cm.\n",
+    "\n",
+    "<img src=\"P-B9.6.png\" width=\"400\">\n",
+    "\n",
+    "[This problem is a slightly modified version from OpenStax. Access for free](https://openstax.org/books/university-physics-volume-1/pages/10-problems)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# DIY Cell"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {
+    "tags": [
+     "hide-input",
+     "hide-output"
+    ]
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Mass: 1.66666666666667 kg\n"
+     ]
+    }
+   ],
+   "source": [
+    "%reset -f\n",
+    "\n",
+    "#this is a static problem\n",
+    "#all linear and angular accelerations must be zero\n",
+    "\n",
+    "import numpy as np\n",
+    "import sympy as sp\n",
+    "\n",
+    "m1 = 5.0\n",
+    "r1 = 0.20\n",
+    "r2 = 0.30\n",
+    "\n",
+    "phi = np.radians(30.0)\n",
+    "\n",
+    "g = 9.81\n",
+    "\n",
+    "M, ft1, ft2 = sp.symbols('M, ft1, ft2')\n",
+    "\n",
+    "#block on incline plane\n",
+    "#balance forces along incline\n",
+    "eq1 = ft1 - m1*g*sp.sin(phi)\n",
+    " \n",
+    "#pulley\n",
+    "#balance torques\n",
+    "eq2 = r2*ft2 - r1*ft1\n",
+    "\n",
+    "#hanging mass\n",
+    "eq3 = M*g - ft2\n",
+    "\n",
+    "sol = sp.solve((eq1,eq2,eq3),(M, ft1, ft2))\n",
+    "print('Mass: '+str(sol[M])+' kg')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<hr style=\"height:2px;border-width:0;color:gray;background-color:gray\">"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<h2>Problem B9.7</h2>\n",
+    "    </header>\n",
+    "\n",
+    "The cart shown below moves across the table top as the block falls. What is the acceleration of the cart? Neglect friction and assume the following data: $m_1 = 2.0$ kg, $m_2 = 4.0$ kg, $I = 0.4$ kgm$^2$, $r = 0.20$ m. \n",
+    "\n",
+    "<img src=\"P-B9.7.png\" width=\"400\">\n",
+    "\n",
+    "[This problem is a slightly modified version from OpenStax. Access for free](https://openstax.org/books/university-physics-volume-1/pages/10-problems)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# DIY Cell"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "metadata": {
+    "tags": [
+     "hide-input",
+     "hide-output"
+    ]
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Acceleration: 1.22625000000000 m/s²\n"
+     ]
+    }
+   ],
+   "source": [
+    "%reset -f\n",
+    "\n",
+    "#Application of N2\n",
+    "\n",
+    "import sympy as sp\n",
+    "\n",
+    "m1 = 2.0\n",
+    "m2 = 4.0\n",
+    "I = 0.4\n",
+    "r = 0.20\n",
+    "\n",
+    "g = 9.81\n",
+    "\n",
+    "a1, a2, alpha, ft1, ft2 = sp.symbols('a1,a2,alpha,ft1,ft2')\n",
+    "\n",
+    "#cart\n",
+    "#horizontal N2\n",
+    "eq1 = m2*a2 - ft2\n",
+    "\n",
+    "#pulley\n",
+    "# + CW\n",
+    "eq2 = I*alpha - r*ft1 + r*ft2 \n",
+    "\n",
+    "#hanging mss\n",
+    "eq3 = m1*a1 - m1*g + ft1\n",
+    "\n",
+    "#equation of constraint leads to same accelerations for the two masses\n",
+    "eq4 = a2 - a1\n",
+    "\n",
+    "#no slip\n",
+    "eq5 = a1 - r*alpha\n",
+    "\n",
+    "sol = sp.solve((eq1,eq2,eq3,eq4,eq5),(a1, a2, alpha, ft1, ft2))\n",
+    "\n",
+    "print('Acceleration: '+str(sol[a2])+' m/s\\u00b2')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<hr style=\"height:2px;border-width:0;color:gray;background-color:gray\">"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<h2>Problem B9.8</h2>\n",
+    "    </header>\n",
+    "\n",
+    "A uniform 40.0 kg scaffold of length 6.0 m is supported by two light cables, as shown below. An 80.0 kg painter stands 1.0 m from the left end of the scaffold, and his painting equipment is 1.5 m from the right end. If the tension in the left cable is twice that in the right cable, find the tensions in the cables and the mass of the equipment.\n",
+    "\n",
+    "<img src=\"P-B9.8.png\" width=\"400\">\n",
+    "\n",
+    "[This problem is a slightly modified version from OpenStax. Access for free](https://openstax.org/books/university-physics-volume-1/pages/12-problems)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# DIY Cell"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "metadata": {
+    "tags": [
+     "hide-input",
+     "hide-output"
+    ]
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Mass of equipment: 16.0000000000000 kg\n"
+     ]
+    }
+   ],
+   "source": [
+    "%reset -f\n",
+    "\n",
+    "#static problem\n",
+    "\n",
+    "import sympy as sp\n",
+    "\n",
+    "ms = 40.0\n",
+    "mp = 80.0\n",
+    "\n",
+    "L = 6.0\n",
+    "\n",
+    "#let origin be at the left point of the scaffold\n",
+    "rp = 1.0\n",
+    "re = L - 1.5\n",
+    "\n",
+    "g = 9.81\n",
+    "\n",
+    "ft1, ft2, me = sp.symbols('ft1,ft2,me')\n",
+    "\n",
+    "#no horizontal forces\n",
+    "\n",
+    "#vertical forces must balance\n",
+    "fgs = ms*g\n",
+    "fgp = mp*g\n",
+    "fge = me*g\n",
+    "eq1 = ft1 + ft2 -fgs - fgp - fge\n",
+    "\n",
+    "#torques about origin must balance\n",
+    "eq2 = L*ft2 - rp*fgp - (L/2)*fgs - re*fge\n",
+    "\n",
+    "# info about the two tension forces: rigth tension twice of left tension\n",
+    "eq3 = ft1 - 2*ft2\n",
+    "\n",
+    "sol = sp.solve((eq1,eq2,eq3),(ft1, ft2, me))\n",
+    "\n",
+    "print('Mass of equipment: '+str(sol[me])+' kg')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<hr style=\"height:2px;border-width:0;color:gray;background-color:gray\">"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<h2>Problem B9.9</h2>\n",
+    "    </header>\n",
+    "\n",
+    "The uniform boom shown below weighs 700.0 N, and the object hanging from its right end weighs 400.0 N. The boom is supported by a light cable and by a hinge at the wall. Calculate the tension in the cable and the force on the hinge on the boom. Does the force on the hinge act along the boom?\n",
+    "\n",
+    "<img src=\"P-B9.9.png\" width=\"200\">\n",
+    "\n",
+    "[This problem is a slightly modified version from OpenStax. Access for free](https://openstax.org/books/university-physics-volume-1/pages/12-problems)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# DIY Cell"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 32,
+   "metadata": {
+    "tags": [
+     "hide-input",
+     "hide-output"
+    ]
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Tension: 777.627066290374 N\n",
+      " \n",
+      "Reaction force: 777.8174826072636 N, directed at 45.01402477644346 degrees\n",
+      " \n",
+      "The reaction force is not along the boom.\n"
+     ]
+    }
+   ],
+   "source": [
+    "%reset -f\n",
+    "\n",
+    "import numpy as np\n",
+    "import sympy as sp\n",
+    "\n",
+    "fgb = 700.0\n",
+    "fgo = 400.0\n",
+    "\n",
+    "phi1 = np.radians(45)\n",
+    "phi2 = np.radians(20)\n",
+    "\n",
+    "#unknown is the reaction force, fr, along the boom \n",
+    "#(mixture of normal, friction, tension and what else is at the attachment)\n",
+    "#ft is tension force\n",
+    "#L is length of boom\n",
+    "frx,fry,ft,L = sp.symbols('frx,fry,ft,L')\n",
+    "\n",
+    "#horizontal forces must balance\n",
+    "eq1 = frx - ft*sp.cos(phi1)\n",
+    "\n",
+    "#vertical forces must balance\n",
+    "eq2 = fry + ft*sp.sin(phi1) - fgb - fgo\n",
+    "\n",
+    "#let origin be at the attachment\n",
+    "#torques about the origin must balance\n",
+    "eq3 = L*ft*sp.sin(sp.pi - phi1 - phi2) - L*fgo*sp.sin(sp.pi/2 + phi2) - (L/2)*fgb*sp.sin(sp.pi/2 + phi2)\n",
+    "\n",
+    "sol = sp.solve((eq1,eq2,eq3),(frx,fry,ft,L))\n",
+    "\n",
+    "frx = float(sol[0][0])\n",
+    "fry = float(sol[0][1])\n",
+    "ft = sol[0][2]\n",
+    "\n",
+    "fr = np.sqrt(frx**2 + fry**2)\n",
+    "theta = np.arctan(fry/frx)*180/np.pi\n",
+    "\n",
+    "print('Tension: '+str(ft)+' N')\n",
+    "print(' ')\n",
+    "print('Reaction force: '+str(fr)+' N, directed at '+str(theta)+' degrees')\n",
+    "print(' ')\n",
+    "print('The reaction force is not along the boom.')"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},