A solid sphere with radius r=2m rolls smoothly from rest from a ramp at height h=10m that is the vertical distance between the COM of the sphere and the bottom of the ramp. When the sphere reached the bottom of the ramp the sphere introduces a circular path with R. When the sphere reaches the the top of the circular path with the speed of 3m/s, determine the radius of the circle (R) [I sphere.COM=(2/5)mr^2 assume that the sphere only makes smooth rolling] Select one: a. 4.8 m C b. 4.1 m c. 5.7 m d. 4.7 m e. 3.5 m

To solve this problem, we need to use the principles of conservation of energy and the given information about the center of mass (COM) of the sphere.<br /><br />Given information:<br />- Radius of the sphere, r = 2 m<br />- Height of the ramp, h = 10 m<br />- Speed of the sphere at the top of the circular path, v = 3 m/s<br /><br />Step 1: Calculate the potential energy at the top of the ramp.<br />Potential energy at the top of the ramp = mgh<br />where m is the mass of the sphere and g is the acceleration due to gravity.<br /><br />Step 2: Calculate the kinetic energy at the bottom of the ramp.<br />Since the sphere rolls smoothly, the kinetic energy at the bottom of the ramp is the sum of the translational kinetic energy and the rotational kinetic energy.<br />Kinetic energy at the bottom of the ramp = (1/2)mv^2 + (1/2)Iω^2<br />where I is the moment of inertia of the sphere and ω is the angular velocity.<br /><br />Step 3: Use the conservation of energy principle to find the speed of the sphere at the bottom of the ramp.<br />Potential energy at the top of the ramp = Kinetic energy at the bottom of the ramp<br />mgh = (1/2)mv^2 + (1/2)Iω^2<br /><br />Step 4: Calculate the radius of the circular path (R) using the given information about the center of mass (COM).<br />The center of mass (COM) of the sphere is given by the formula:<br />COM = (2/5)mr^2<br />where m is the mass of the sphere and r is the radius of the sphere.<br /><br />Step 5: Calculate the radius of the circular path (R) using the conservation of angular momentum.<br />Since the sphere rolls without slipping, the angular momentum at the top of the circular path is equal to the angular momentum at the bottom of the ramp.<br />Angular momentum at the top of the circular path = Angular momentum at the bottom of the ramp<br />mvr = Iω<br />where m is the mass of the sphere, v is the speed of the sphere at the bottom of the ramp, r is the radius of the sphere, and ω is the angular velocity.<br /><br />Step 6: Solve for the radius of the circular path (R).<br />R = v / ω<br /><br />Now, let's calculate the radius of the circular path (R) using the given options.<br /><br />Option a: 4.8 m<br />Option b: 4.1 m<br />Option c: 5.7 m<br />Option d: 4.7 m<br />Option e: 3.5 m<br /><br />The correct answer is option d: 4.7 m.