a) The gravitational potential energy U(r)of a mass mdue to a mass density ρ(r)satisfies ∇^2 U= 4πGmρ, whereGis the gravitational constant. If the earth is considered to be a uniform sphere of mass M, radius R, show that thegravitational potential energy of a mass minsidethe earth a distance rfrom the center is
U(r)=mg/2R(r2-3R2)
whereg= GM/R2= 9.81 m/s^2.
b) A tunnel is to be constructed through the earth, along which a frictionless train will run between cities at A and B.The track is described by the curve r(Φ)(polar coordinates).
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIUAAACGCAIAAAB8LN7FAAAUmElEQVR4nO2daXBT1d/HdcZxBhwUoeCUxSIwDDI6sugMDgKyjY6ICij4AqaOLPoCcEa2IijIosIIqANoN2jTpEmzJ02zNQlNyE33NCFpuiVtmmZpmpKkTcjSNLnPi8y/w0Nubm/W3pR+XnVuzz33JN/k5Jzfdp4Dp0ATz030AKb4f0zpgS6m9EAXU3qgiyk90MWUHugiY/Tw+Xw2m02j0Wg0mubmZrFYXFZWlp+fT6fT+Xw+j8fj8/lcLreoqKioqIjH49XV1alUKo1Go9frXS5XKBSa6FeACPTqMTIyYrVaFQpFbW1tRUUFi8Wqrq7G4XAAALS2tnZ1dXV1dWm1WqvV6nA47Ha73W5/9OhRd3e3VqvV6XQdHR0tLS0MBoNEIgmFQhqNJhKJGhoatFrt48ePJ/rFRQV1ejidzocPH4pEIiaTWV1dLRQK29vbzWazy+WKu8+RkZFHjx719vZKpVI+n89isfh8PgAAJpMpEAgkcfCJgxY9XC6XUqkUCASVlZUCgUClUjkcjmAwmIpneTweo9EolUq5XC6bzQYAwGg0puJBcTDxeuj1ej6fT6FQhEJhT0+Pz+dL26NDoZDNZmtubiaTyXQ6XS6XezyetD0dkgnTIxQKtbe3k8lkGo2mVqsn/I0wmUw8Hg+LxUokkuHh4YkaxgToMTo62traWlFRwWAwent70z8AGNxuNwAAZWVlEonE4XCkfwDp1qOjo4NKpTKZTLQp8SQej6e+vp5AIAAAkMg6Ig7Sp4fFYqFSqXQ6Xa/Xp+2hieByuQAAIBKJzc3NaXtozHrEsbEaHR2VSCREIlGpVMZ674Rjs9mqqqrKy8vjWIPZ7XaHw2GxWGw2G8JbYtDD4XD88ccfsS7YjUYjiUQSiUQT/oudCBqNBofDyWSymO6SSCQ7duz45Zdfjh8/zuVykXyUY9CjpKRkwYIFDx8+RH5LfX09BoPR6XTIb0EtXq+Xz+djsdhHjx4hvMVkMq1Zs0alUg0NDa1YsQLJAgGpHi6X69atW7t378bhcEjau91uGo3G4XC8Xi/CR2QESqUSi8Wq1WokjbFY7ObNm/v6+vLz83/88UckWyukeuBwuPr6+v379+Px+HEbGwwGPB7f0NCAsPPMwmq1YjCYBw8ejNvy4MGD27Zt++mnn7KysgYGBpB0jkgPjUaTl5eHwWD27Nnz3XffwZsx1Go1BoPp6elB0nOGMjIywmazGQwGzEfe7XZv3LgRg8F4PJ7jx4//888/SHoeXw+Px1NYWDgwMBAIBKRS6WeffQbTuKmpiUAgDA4Owowy/MeE7LaSi0QiIRAI0Tbzcrl84cKF4c/lV199de/ePSR9jqNHMBi8du3a7t27/X6/y+U6f/78+vXrRSIRZGOpVEoikaL9YMjl8osXL4Zn3qGhobNnz0brJ4NoamrC4/GRe0aPx5Obm7tmzZri4uLTp08fP37c7/cj6XAcPUKhkNVqNZvNo6OjIyMjBoPBYrFYrdbIllKplMViwUxlFoslJycnPPQbN24QicSSkhIkQ0Q5bW1tFRUVT31LRkdHzWazzWbr6uqKyRKRnP05AABsNjva+joYDDY2NpJIpGPHjgUCAbPZfOnSpcLCQoPBkJSnTzhtbW0kEmloaCjxrpKgh0KhEAgEo6OjkP91OBxXrlwhkUirVq26evUqCIICgSAvL49CoST+aPTQ2dnJZrMRTkowJKqHWCzeuXNnV1dXtAYYDOa///4DQXDLli0KhQIEwaNHj54/fz7xoaONM2fO5OXlJehwTEgPnU537ty5ZcuWLV++vLq6OrKByWQ6dOiQz+e7f//+zp07nU5nV1fX2rVrRSJRIBAYGRlJ5Onowel0Hjp0aNq0abm5uVVVVYl0Fb8eDoejuLh4cHCwr69vx44dzz///IULF55aXCkUio0bNzY2Nq5cuXL37t0sFquuro5CoeBwOAqF4nQ6Exk6SqipqXn77bfnzZtHIpFAEORwOEi2itGIUw+/319WVjZmmAoGgzdv3pwxY8amTZue3JYHg0GBQIDH46urq0tLS8d+8fh8Pnpc1nHjcrl+/fXX6dOnb9myRavVhi+OjIzcvXtXo9HE12ecenA4nPv37z91sba29v333589e/Zvv/02tu+brMhksk2bNmVlZV27du2pXbrdbr93757dbo+j23j0UKlUNBoN8l9DQ0MXLlx49dVX169fDwBAHJ2jH7vdnpeXN3PmzK1bt0az0bW1tWGx2Dh8RTHrMTg4iMVi4af+urq6Dz/8cPr06T/88EN/f//YdYvFkmb3Z9KhUqlvvfXWnDlz/vzzT3jTtUgkitVfAsaqRygUolKpSNx8gUDg1q1bCxYsWLJkSUFBQXh1i8Viw/tYr9ebcetduVz+xRdfvPDCC3v27Ons7By3vc/nw+PxFoslpqfEpodara6srETe3mAwHD58eNq0aevWrbt27ZpUKgVBMBgMcrncDLIn6nS6I0eOvPzyy2vWrGEwGMhv7O7uptFoMe1IYtDD4/GUl5fD2G6jEbYKP/fcc5999plIJFKr1S0tLbF2MiFotdozZ87MmzdvyZIlf/31VxyLFC6XG94FIyQGPaRSafgDHgcNDQ03btzYvn37rFmz1q5di8ViUT5ftbW1nTx5Mjs7e8GCBZcuXYp12hnD6XRWVFQgj+BGqofT6SwpKYnvTRwaGiovLw8EAqFQ6Pr16y+++OL06dPXrVtXXFyMwlmrvr7+4MGDWVlZS5cuPX/+fOKONZlMhnyHiFQPOp3e1NQU34B0Op1cLgdBMBgMCoVCEom0b9++zz//fMaMGYsXLz516lRM3+gUYbPZcDjcxo0bX3rppZUrV/79998IPazj4vP5YNxWT4FIj8HBQRKJFHdKy5hTJBzxD4KgTCaj0+lqtfrcuXOLFy+eMWPG5s2b79y5g2TdklzsdjubzT506FB2dvbMmTO3b9/OYDCSblhTKpV8Ph9JS0R60On0uA0AT+L3+8fM8rW1tVwuFwTB4eFhEom0a9eu+fPnZ2VlffLJJzdv3pTJZKkLag6FQjqdjkgkHj58+M0335w1a9a777575coVhFEjcTA6OkogEJA4SMbXY3BwsLS0NBV5KwAAPLnP7+npKSws/Prrr5cvXz579uzVq1d/++23//77LwAAFoslwVyQ4eHhsNfo3LlzGzZsyMnJyc7O/uCDD37++efa2to0JDk0NjbW1NSM22x8PWQyWermd5lMRqVSn1omWCwWNpt99uzZbdu2LV26NCsra9myZe+9996BAwcuX75cUlIikUhUKpVer7dYLE6n0/s/3G532Lvc2dnZ2NhIp9Nv3rx54sSJjz/+eMWKFYsWLZozZ87atWtzc3Pv3r2rUqnSafD3eDwwwQVjjKOH1+ulUCgpNQ6GJYn21pjNZolEkp+ff/jw4U8//XTVqlXTp0+fO3fuggULFi5cmJOTM3/+/Llz57722mtz587Nzs7OycnJycl5/fXXs7OzX3nllUWLFm3evHnv3r0XL15kMBgajWYCw1bFYvG4po1x9NBoNEKhMHlDggZekifx+Xx9fX1KpVIikTAYDAKBcOPGjZMnT54+ffrUqVMXL17EYrFEIlEgEDQ0NHR1daHKxdLf30+j0aI5tsPA6REKhVgsVnqiDgAAoFAok8ZjCEkwGKRQKCaTCaYNnB52u51MJqftPZLJZBQKBeX79gRRKBTwRl84PRQKBZIlQRJpaGig0+npTOlMMw6Hg8FgwCxW4fRgMBjwX65kEQwGxxYeYrE4csU1aQiFQvDvalQ93G532maPsM/5/Pnz33//fVFRkUKhmMQTFwAAjY2N0f4bVQ+dTodwi58UysvLN2zY8ODBA5VKBYJgbW0tEkkCgYDT6TSbzUaj0Wg02mw29GdhWa1WmJigqHqIxeKk2EgQcvjw4aciFmtra8lkMqQkjx8/VqvV9+/fr6ioEAqF4ieg0+ksFqu+vh4yyBgN+P1+FosV7XMTVQ8SiYQ8MStBOjo61q9fHxkBVFdXRyaTwz/vYWum3+8Xi8U4HI7P57e3t9vt9ieXf6FQyO12GwwGiURCo9GEQuGT3nv0wOPxou0ioPV4/PgxnU5PUfmQSOrr6wsLC6P9i0qlDgwMNDc39/b2EgiE6upqJOmqfr9fpVJVVFSgMMxFLpdHS8OE1sNoNKIn3rmxsbGgoODo0aO3b9+GCRSG5PHjxywWi06no2qn2dfXF+3thdajtbUVDT6iMWQy2TfffFNUVNTe3g7fMhAIRBrq6+rqmEwmerY1w8PD0aYfaD2YTCbakjOUSiWZTB7Xsmm1WsvLyyOvy2SyysrKtM3A8ASDQTabDflaoPUgEolxxJGkmoaGhnEXwTU1NSdOnIC02TEYjLDbGA2UlZWZzebI6xB6BAIBgUCAzrxxAAAwGEy0maegoOD3338/duzY2bNnOzo6nvqv2+3G4/FJyWJKnPv373d3d0deh9DD7XZzuVyUfLUjEQgEGAwm8lvS3Nx87NgxnU5XXl5+5cqVvXv3Rr6ExsZGlCSRtrW1hXe+TwGhh9Vqzc/PT/2Q4qe6ujryW3L79u0jR44MDQ3hcDi5XL5169ZIs53f7y8tLUXDV1+v10MGs0HoYTabo4Wvo4fq6uqysrInJWlpafnoo484HA4Gg6mpqbl16xZkQAyLxUqn3SEaBoOhrKws8jqEHiaTKe44xHQS/pY8OXERicQDBw7k5ubCWN4MBgORSEzLAOEwm813796NvA6hR3d3d9yhb2km8reks7Pz+vXrMKFibrc7naaHaDidTg6HE3kdQg+tVhtTUaWJpaamprKyckyAwcFBAoEAf0s4mzT1Q4PD6/UKBILI69B6ZFYhNwAAKioqOjo6tFpte3u7XC7XwlJYWKhUKuHbpJpw7ezI1zIZ9ABBkE6n3759u6CgoLCwsLi4uACW27dv5+fnw7dJNTdu3CgtLY18IRk/X2UooVAIcicErQd67AqTlaGhIcgKChB6GI1GFPoM0s/AwACFQsHj8Xg8PukJXVarFbLSHvT+g81mJ/fxmYjb7T558uTWrVsfPHiwb9+++vr6JHbe29tbUFAQeR1Cj/7+/mjeumeN3bt3FxUVgSCYm5t7586dJPbc09MDGYgLocfw8DCTyZzwHdOE09raunr1ajwef/Xq1QsXLiQ3qLy9vR1y0QShx8jIiFAonKzhT8hhMBhffvnl5cuXd+3alfTOAQCALEsM7Y9iMBgozLRMM2fOnCkpKfH5fO+8807SvdcEAgGpPwoEQSKRiObzB9JAZ2fnG2+8ETa97N+//+DBg0ms9BEKhaqqqiAT8qD1aGlpaW1tTdbjMxE+n3/y5EkmkwmCoEQiuXjx4rihFMgJR71AOpWjxvug3wWSuVgslnDxskig9RgeHqbRaJlyZl/GoVaro50pAq1HKBQikUioSvaaTLBYrGhnPkSN3xUIBJABEFMkSCAQYLFY0cJcourR2dnJ4/FSNqpnF5vNRiQSo/0WRNXD6XTS6XT4ZNAp4kCpVMK4w8fJV0NtFkXmApNsAMLr0dLSks6jxZ4FhoeHKyoqYGxRcHoMDAygJwZ5cqDRaOBTluH0CAaDTCYz7tJoU0TC4/HgDVHj1MtoamoSi8VJHdKzi91uH7csyDh6uFwuIpGIhoDXSUBtbe24jvDx6y1xOBzISOwpYiIQCJBIpHGLrI2vh8lkmrItJk57e3u4Hh48SOv1PePukAQJhUL37t1DsjJCpIder0dDTHjm0traivANRFrvlUwmT+4jHlPH6OhoYWEhwiOekeoxMDBAIBCm9oZxIJfLWSwWwsYx1Avn8XioSkrPCDweDx6PRx4cEoMe4VLkk/7gm+Ty4MGDmE4Bie28icbGRsgskikgCU/yMe2mY9MjEAhMhQIhJBgM4nC4WAuuxHxekclkKi8vR08tENQCAABkhiA88ZznJZVK01k6LhMxm83FxcVxFKuLR49QKEQmk5/xgDkYvF5vaWlpfAcsxnn+oMvlwmKxyTohY5JRVVUFU7ESnvjPS9Xr9SUlJVPL36eI72djjITOE1YoFPDVfZ81wnaqRFI1knDediIfh8lEb28vHo9P8BiZRPUIBoMcDgcyVfSZwmg0YrHYxMOjEtUDBMFgMIjH45/lfbvNZispKUmK/TsJeoAgGAgEyGQylUpNSm+Zhclkunfvnl6vT0pvydEDBMFgMIjFYpG4JCcTBoMhiWKASdQjDJfLZbPZz4g1RaPR4PH45G7CkqwH+L/Dh9JWa3yiqKmpIZPJSc96Tb4eIAiqVCocDjdZ/bter7eqqopCoSA/lRY5KdEDBEGLxUIkEpGf25opmEwmLBZbW1ubov5TpQcIgj6fr7Kykkql2u321D0lndTV1eFwuJSmjaVQjzAKhQKHw2XKgefRsFqtFAqFxWKlYo56kpTrAYKg3W5nsVgMBgOyIgHK8fv9MpmMQCCkx7+QDj3CtLW14XA4gUCQKWm7oVBIqVQSiUSRSJQ2M3b69ABB0Ov11tfXk0ikmpqaJJafSAVdXV0UCoXJZPb19aXzuWnVI8zQ0JBEIikrKxOLxShUpa2tjUKhcDicCVmvT4AeYdxut1QqDZ81G59rM+njkcvlOByOTqdP4M5pwvQI4/P55HI5lUoNH86R/l291+vVarUCgQCPx4tEogn/ZEywHmP09fWFT+Zks9lyudxsNqe0IJrD4WhvbxcKhWQymc1mq9VqlMycaNEjjM/n0+v1dXV1NBqNxWLxeDyVSmU0GhNf3oyMjAwODnZ3d4tEIg6HQyKRRCLRw4cP0VZ2DV16jBEMBh0Oh1qtbmhoIJFIlZWVXC6XyWQ2Nzd3dXXpdDqTyeR0Or1er9/v9z2B3+93uVw2m02n02m12tbWVqFQSKfThUIhlUoViURNTU0GgwG1FmiU6vEUPp+vv78/fKYMAAA4HI5AIPB4PKFQKIqgurqaTqfn5+dzOJzm5maNRtPT0zM8PJwR5aMyQ49o+Hw+bwQZXXMls/WYfEzpgS6m9EAXU3qgiyk90MWUHuji/wAuegnvUvZreQAAAABJRU5ErkJggg==)
Use energy conservation to derive an expression for the speed v of a train which starts at rest at A as it passes throughsegment dl of the curve at r(Φ), and hence show that the journey time is
T = ∫dt = ∫dl/v = (√R/g ΦA∫Φg √(r2+r'2)/R2-r2) dΦ.
where r' = dr/dΦ.
c) The tunnel is to be constructed so as to minimize the journey time. Taking care to note the nature of the integrand,write down an Euler-Lagrange equation for the extremal curve r(Φ) [not asked to solve this].
Let α be the minimum distance of the tunnel from the center of the earth, where dr/dΦ = 0. Use this to obtain anexpression for dr/dΦ in terms of r,R, and α.
d) Notingd? = dr/(dr/dΦ), reexpressT as an integral
T = 2a∫R..dr
and evaluate.
e) If R = 6,400 km determine the journey time in minutes through the center of the earth.