Merge branch 'master' into collaboration-sarah

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diff --git a/src/client/views/nodes/PhysicsBox/PhysicsSimulationQuestions.json b/src/client/views/nodes/PhysicsBox/PhysicsSimulationQuestions.json
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+{
+  "inclinePlane": [
+    {
+      "questionSetup": [
+        "There is a 1kg weight on an inclined plane. The plane is at a ",
+        " angle from the ground. The system is in equilibrium (the net force on the weight is 0)."
+      ],
+      "variablesForQuestionSetup": ["theta - max 45"],
+      "question": "What are the magnitudes and directions of the forces acting on the weight?",
+      "answerParts": [
+        "force of gravity",
+        "angle of gravity",
+        "normal force",
+        "angle of normal force",
+        "force of static friction",
+        "angle of static friction"
+      ],
+      "answerSolutionDescriptions": [
+        "9.81",
+        "270",
+        "solve normal force magnitude from wedge angle",
+        "solve normal force angle from wedge angle",
+        "solve static force magnitude from wedge angle given equilibrium",
+        "solve static force angle from wedge angle given equilibrium"
+      ],
+      "goal": "noMovement",
+      "hints": [
+        {
+          "description": "Direction of Force of Gravity",
+          "content": "The force of gravity acts in the negative y direction: 3π/2 rad."
+        },
+        {
+          "description": "Direction of Normal Force",
+          "content": "The normal force acts in the direction perpendicular to the incline plane: π/2-θ rad, where θ is the angle of the incline plane."
+        },
+        {
+          "description": "Direction of Force of Friction",
+          "content": "The force of friction acts in the direction along the incline plane: π-θ rad, where θ is the angle of the incline plane."
+        },
+        {
+          "description": "Magnitude of Force of Gravity",
+          "content": "The magnitude of the force of gravity is approximately 9.81."
+        },
+        {
+          "description": "Magnitude of Normal Force",
+          "content": "The magnitude of the normal force is equal to m*g*cos(θ), where θ is the angle of the incline plane."
+        },
+        {
+          "description": "Net Force in Equilibrium",
+          "content": "For the system to be in equilibrium, the sum of the x components of all forces must equal 0, and the sum of the y components of all forces must equal 0."
+        },
+        {
+          "description": "X Component of Normal Force",
+          "content": "The x component of the normal force is equal to m*g*cos(θ)*cos(π/2-θ), where θ is the angle of the incline plane."
+        },
+        {
+          "description": "X Component of Force of Friction",
+          "content": "Since the net force in the x direction must be 0, we know the magnitude of the x component of the friction force is m*g*cos(θ)*cos(π/2-θ)."
+        },
+        {
+          "description": "Y Component of Normal Force",
+          "content": "The y component of the normal force is equal to m*g*cos(θ)*sin(π/2-θ), where θ is the angle of the incline plane. The y component of gravity is equal to m*g"
+        },
+        {
+          "description": "Y Component of Force of Friction",
+          "content": "Since the net force in the x direction must be 0, we know the magnitude of the y component of the friction force is m*g-m*g*cos(θ)*sin(π/2-θ)."
+        },
+        {
+          "description": "Magnitude of Force of Friction",
+          "content": "Combining the x and y components of the friction force, we get the magnitude of the friction force is equal to sqrt((m*g*cos(θ)*cos(π/2-θ))^2 + (m*g-m*g*cos(θ)*sin(π/2-θ))^2)."
+        }
+      ]
+    },
+    {
+      "questionSetup": [
+        "There is a 1kg weight on an inclined plane. The plane is at a ",
+        " angle from the ground. The system is in equilibrium (the net force on the weight is 0)."
+      ],
+      "variablesForQuestionSetup": ["theta - max 45"],
+      "question": "What is the minimum coefficient of static friction?",
+      "answerParts": ["coefficient of static friction"],
+      "answerSolutionDescriptions": [
+        "solve minimum static coefficient from wedge angle given equilibrium"
+      ],
+      "goal": "noMovement",
+      "hints": [
+        {
+          "description": "Net Force in Equilibrium",
+          "content": "If the system is in equilibrium, the sum of the x components of all forces must equal 0. In this system, the normal force and force of static friction have non-zero x components."
+        },
+        {
+          "description": "X Component of Normal Force",
+          "content": "The x component of the normal force is equal to m*g*cos(θ)*cos(π/2-θ), where θ is the angle of the incline plane."
+        },
+        {
+          "description": "X Component of Force of Friction",
+          "content": "The x component of the force of static friction is equal to μ*m*g*cos(θ)*cos(π-θ), where θ is the angle of the incline plane."
+        },
+        {
+          "description": "Equation to Solve for Minimum Coefficient of Static Friction",
+          "content": "Since the net force in the x direction must be 0, we can solve the equation 0=m*g*cos(θ)*cos(π/2-θ)+μ*m*g*cos(θ)*cos(π-θ) for μ to find the minimum coefficient of static friction such that the system stays in equilibrium."
+        }
+      ]
+    },
+    {
+      "questionSetup": [
+        "There is a 1kg weight on an inclined plane. The coefficient of static friction is ",
+        ". The system is in equilibrium (the net force on the weight is 0)."
+      ],
+      "variablesForQuestionSetup": ["coefficient of static friction"],
+      "question": "What is the maximum angle of the plane from the ground?",
+      "answerParts": ["wedge angle"],
+      "answerSolutionDescriptions": [
+        "solve maximum wedge angle from coefficient of static friction given equilibrium"
+      ],
+      "goal": "noMovement",
+      "hints": [
+        {
+          "description": "Net Force in Equilibrium",
+          "content": "If the system is in equilibrium, the sum of the x components of all forces must equal 0. In this system, the normal force and force of static friction have non-zero x components."
+        },
+        {
+          "description": "X Component of Normal Force",
+          "content": "The x component of the normal force is equal to m*g*cos(θ)*cos(π/2-θ), where θ is the angle of the incline plane."
+        },
+        {
+          "description": "X Component of Force of Friction",
+          "content": "The x component of the force of static friction is equal to μ*m*g*cos(θ)*cos(π-θ), where θ is the angle of the incline plane."
+        },
+        {
+          "description": "Equation to Solve for Maximum Wedge Angle",
+          "content": "Since the net force in the x direction must be 0, we can solve the equation 0=m*g*cos(θ)*cos(π/2-θ)+μ*m*g*cos(θ)*cos(π-θ) for θ to find the maximum wedge angle such that the system stays in equilibrium."
+        },
+        {
+          "description": "Simplifying Equation to Solve for Maximum Wedge Angle",
+          "content": "Simplifying 0=m*g*cos(θ)*cos(π/2-θ)+μ*m*g*cos(θ)*cos(π-θ), we get cos(π/2-θ)=-μ*cos(π-θ)."
+        },
+        {
+          "description": "Simplifying Equation to Solve for Maximum Wedge Angle",
+          "content": "The cosine subtraction formula states that cos(A-B)=cos(A)*cos(B)+sin(A)sin(B)."
+        },
+        {
+          "description": "Simplifying Equation to Solve for Maximum Wedge Angle",
+          "content": "Applying the cosine subtraction formula to cos(π/2-θ)=-μ*cos(π-θ), we get cos(π/2)*cos(θ)+sin(π/2)*sin(θ)=-μ*(cos(π)cos(θ)+sin(π)sin(θ))."
+        },
+        {
+          "description": "Simplifying Equation to Solve for Maximum Wedge Angle",
+          "content": "Simplifying cos(π/2)*cos(θ)-sin(π/2)*sin(θ)=-μ*(cos(π)cos(θ)-sin(π)sin(θ)), we get -sin(θ)=-μ*(-cos(θ))."
+        },
+        {
+          "description": "Simplifying Equation to Solve for Maximum Wedge Angle",
+          "content": "Simplifying -sin(θ)=-μ*(-cos(θ)), we get tan(θ)=-μ."
+        },
+        {
+          "description": "Simplifying Equation to Solve for Maximum Wedge Angle",
+          "content": "Solving for θ, we get θ = atan(μ)."
+        }
+      ]
+    }
+  ]
+}