A uniform solid sphere of mass M and radius R is rolling without sliding along a level plane with a speed v = 3.15 m/s when it encounters a ramp that is at an angle θ = 15.10° above the horizontal. Find the maximum distance that the sphere travels up the ramp in each case. The ramp is frictionless, so the sphere continues to rotate with its initial angular speed until it reaches its maximum height.
Question
A uniform solid sphere of mass M and radius R is rolling without sliding along a level plane with a speed v = 3.15 m/s when it encounters a ramp that is at an angle θ = 15.10° above the horizontal. Find the maximum distance that the sphere travels up the ramp in each case. The ramp is frictionless, so the sphere continues to rotate with its initial angular speed until it reaches its maximum height.
Solution
To solve this problem, we need to use the principles of conservation of energy and the relationship between linear and angular velocity for a rolling object.
Step 1: Identify the initial and final conditions The sphere starts with a certain kinetic energy due to its linear and rotational motion. When it reaches the maximum height on the ramp, all this kinetic energy will have been converted into potential energy.
Step 2: Calculate the initial kinetic energy The kinetic energy of a rolling object is the sum of its translational kinetic energy (1/2 m v^2) and its rotational kinetic energy (1/2 I ω^2). For a solid sphere, the moment of inertia I = 2/5 m R^2, and the angular velocity ω = v/R. So the total initial kinetic energy K_i = 1/2 m v^2 + 1/2 (2/5 m R^2) (v/R)^2 = 7/10 m v^2.
Step 3: Calculate the final potential energy The potential energy at the maximum height is m g h, where g is the acceleration due to gravity and h is the height. The height can be expressed in terms of the distance d the sphere travels up the ramp: h = d sin θ.
Step 4: Apply conservation of energy The initial kinetic energy equals the final potential energy: 7/10 m v^2 = m g d sin θ. We can solve this equation for d to find the maximum distance the sphere travels up the ramp: d = (7/10) (v^2 / g) / sin θ.
Step 5: Substitute the given values Substitute v = 3.15 m/s, g = 9.81 m/s^2, and θ = 15.10° into the equation to find the maximum distance d.
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