add exercises

notebooks/newtonLawForParticles.ipynb

@@ -799,9 +799,7 @@
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@@ -861,24 +859,30 @@
    "source": [
     "## Problems\n",
     "\n",
-    "1. (Example 13.1 of Hibbeler's book) A 50 kg crate rests on a horizontal surface for which the coefficient of kinetic friction is $\\mu_k$ = 0.3. If the crate is subjected to a 400-N towing force applied upward with a $30^o$ angle w.r.t. the horizontal, determine the velocity of the crate after 3 s starting from rest.  \n",
+    "1. (Example 13.1, Hibbeler's book) A $50 kg$ crate rests on a horizontal surface for which the coefficient of kinetic friction is $\\mu_k = 0.3$. If the crate is subjected to a $400 N$ towing force applied upward with a $30^o$ angle w.r.t. the horizontal, determine the velocity of the crate after $3 s$ starting from rest.  \n",
     "\n",
-    "2. (Example 13.2 of Hibbeler's book) A 10 kg projectile is fired vertically upward from the ground, with an initial velocity of 50 $m/s$. Determine the maximum height to which it will travel if (a) atmospheric resistance is neglected; and (b) atmospheric resistance is measured as $F_D=0.01v^2$ $N$, where $v$ is the speed of the projectile at any instant, measured in $m/s$.  \n",
+    "2. (Example 13.2, Hibbeler's book) A 10 kg projectile is fired vertically upward from the ground, with an initial velocity of 50 $m/s$. Determine the maximum height to which it will travel if (a) atmospheric resistance is neglected; and (b) atmospheric resistance is measured as $F_D=0.01v^2$ $N$, where $v$ is the speed of the projectile at any instant, measured in $m/s$.  \n",
     "\n",
-    "3. (Exercise 13.6 of Hibbeler's book) A person pushes on a 60-kg crate with a force F. The force is always directed down at $30^o$ from the horizontal as shown, and its magnitude is increased until the crate begins to slide. Determine the crate’s initial acceleration if the coefficient of static friction is $\\mu_s=0.6$ and the coefficient of kinetic friction is $\\mu_k=0.3$.  \n",
+    "3. (Exercise 13.6, Hibbeler's book) A person pushes on a $60 kg$ crate with a force $F$. The force is always directed down at $30^o$ from the horizontal as shown, and its magnitude is increased until the crate begins to slide. Determine the crate’s initial acceleration if the coefficient of static friction is $\\mu_s=0.6$ and the coefficient of kinetic friction is $\\mu_k=0.3$.  \n",
     "\n",
-    "4. (Exercise 13.46 of Hibbeler's book) The parachutist of mass $m$ is falling with a velocity of $v_0$ at the instant she opens the parachute. If air resistance is $F_D = Cv^2$, show that her maximum velocity (terminal velocity) during the descent is $v_{max}=\\sqrt{{mg}/{C}}$. Solve the problem numerically considering m=100 $kg$, g=10 $m/s^2$, and C=10 $kg/m$. Plot a simulation of this numerical solution and show that indeed the parachutist approaches the terminal velocity.\n",
+    "4. (Exercise 13.46, Hibbeler's book) The parachutist of mass $m$ is falling with a velocity of $v_0$ at the instant she opens the parachute. If air resistance is $F_D = Cv^2$, show that her maximum velocity (terminal velocity) during the descent is $v_{max}=\\sqrt{{mg}/{C}}$. Solve the problem numerically considering $m=100 kg$, $g=10 m/s^2$, and $C=10 kg/m$. Plot a simulation of this numerical solution and show that indeed the parachutist approaches the terminal velocity.\n",
     "\n",
-    "5. Consider a block with mass of 1 kg attached to a spring hanging from a ceiling (the spring constant k = 100 N/m). At t = 0 s, the spring is stretched by 0.1 m from the equilibrium position of the block + spring system and then it's released (the initial velocity is not specified). Find the motion of the block.\n",
+    "5. Consider a block with mass of $1 kg$ attached to a spring hanging from a ceiling (the spring constant $k = 100 N/m$). At $t = 0 s$, the spring is stretched by $0.1 m$ from the equilibrium position of the block + spring system and then it's released (the initial velocity is not specified). Find the motion of the block.\n",
     "\n",
     "6. Solve exercises 12.1.16, 12.1.19, 12.1.24, 12.1.29, 12.1.30, 12.1.31(a, b, d) and 12.1.32 from Ruina and Pratap's book (2019).  \n",
     "\n",
-    "5. (https://youtu.be/N6IhkTjWrd4) A 15 kg box rests on a frictionless horizontal surface attached to a 5 kg box as shown below.  \n",
+    "7. (https://youtu.be/N6IhkTjWrd4) A $15 kg$ box rests on a frictionless horizontal surface attached to a $5 kg$ box as shown below.  \n",
     "  a. What is the acceleration of the system?  \n",
     "  b. What is the tension in the rope?  \n",
-    "  c. Now consider a coefficient of kinetic friction of 0.25 between the horizontal surface and the box. What are the acceleration of the system and the tension in the rope? \n",
+    "  c. Now consider a coefficient of kinetic friction of $0.25$ between the horizontal surface and the box. What are the acceleration of the system and the tension in the rope? \n",
     "  \n",
-    "<figure><center><img src=\"../images/boxes_pulley_rope.png\" alt=\"free-body diagram of the ball\" width=\"300\"/></center></figure>"
+    "<figure><center><img src=\"../images/boxes_pulley_rope.png\" alt=\"free-body diagram of the ball\" width=\"300\"/></center></figure>\n",
+    "\n",
+    "8. Consider a moving particle in a fluid with viscosity proportional to the cube of the velocity. This particle moves only vertically and never reaches the ground. Consider the magnitude of the acceleration due to gravity as $10 m/s^2$.  \n",
+    "   a. Draw the free body diagram.  \n",
+    "   b. What is the differential equation that describes the particle's motion?  \n",
+    "   c. If the viscosity coefficient is $5 Ns^3/m^3$, what is the maximum speed of the particle?\n",
+    "\n"
    ]
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@@ -918,7 +922,7 @@
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