Calculating Sled Speed on a Frictionless Hill: Equations for Theta and Radius

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To calculate the sled's speed at an angle theta on a frictionless hemispherical hill, conservation of mechanical energy is applied. The sled starts from rest, so its initial potential energy is converted into kinetic energy as it descends. The correct approach involves using the height at angle theta, which can be expressed as h = R(1 - cos(theta)), to determine the speed. Newton's Second Law and centripetal acceleration are then used to analyze the sled's motion at that angle. Setting up the equation involves equating potential energy at the top to kinetic energy at angle theta.
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Homework Statement


A sled starts from rest at the top of the frictionless, hemispherical hill. What is the equation to calculate the sleds speed at angle (theta)

Homework Equations





The Attempt at a Solution



I need to figure out the equation. I'm given theta and the radius. I thought the equation was Rsin(theta) but that was incorrect
 
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We're told that the hill is frictionless, so we can use the conservation of mechanical energy to find the sleds speed at some angle theta. After finding the final speed, you'll want to use Newton's Second Law along with the equation for centripetal acceleration to find the sleds speed at some angle theta. Can you show me how you would set up the equation for the conservation of mechanical energy?
 

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