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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)."
}
]
}
]
}
|
