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 provides enough friction to prevent the sphere from sliding, so both the linear and rotational motion stop (instantaneously).
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 provides enough friction to prevent the sphere from sliding, so both the linear and rotational motion stop (instantaneously).
Solution
To solve this problem, we need to use the principles of conservation of energy and the equations for kinetic energy.
Step 1: Identify the initial and final conditions.
Initially, the sphere is rolling without sliding on a level plane with a speed v. At this point, it has both translational kinetic energy (1/2 Mv^2) and rotational kinetic energy (1/2 Iω^2). Since it's rolling without sliding, we can use the relation v = ωR.
Finally, when the sphere has climbed up the ramp to its maximum height, it momentarily comes to rest. At this point, it has potential energy (Mgh), but no kinetic energy.
Step 2: Set up the conservation of energy equation.
The total initial energy (kinetic + potential) equals the total final energy (kinetic + potential).
1/2 Mv^2 + 1/2 Iω^2 = Mgh
Step 3: Substitute the moment of inertia I and the angular speed ω.
For a solid sphere, I = 2/5 MR^2. And since the sphere is rolling without sliding, ω = v/R.
1/2 Mv^2 + 1/2 * 2/5 MR^2 * (v/R)^2 = Mgh
Step 4: Simplify the equation.
After canceling out the common factors and simplifying, we get:
7/10 v^2 = gh
Step 5: Solve for h, the maximum height the sphere climbs up the ramp.
h = 7/10 * v^2 / g
Substitute the given values v = 3.15 m/s and g = 9.81 m/s^2.
h = 7/10 * (3.15 m/s)^2 / 9.81 m/s^2 = 0.5 m
Step 6: Find the maximum distance the sphere travels up the ramp.
The maximum distance d is related to the maximum height h and the angle of the ramp θ by the relation h = d sin θ.
d = h / sin θ
Substitute the given values h = 0.5 m and θ = 15.10°.
d = 0.5 m / sin 15.10° = 1.9 m
So, the maximum distance that the sphere travels up the ramp is approximately 1.9 m.
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