A uniform solid ball rolls smoothly along a floor, then up a ramp inclined at 13.5 degrees. It momentarily stops when it has rolled 1.65 m along the ramp. What was its initial speed?
How far vertically does it move? What is the final mechanical energy (the sum of U and the two types of K)? What, then, was the initial mechanical energy? What is the rotational inertia (see Table 10-2)? How is the angular speed related to the com speed?
To find the initial speed of the ball, we can use the principle of conservation of mechanical energy. When the ball is at the bottom of the ramp, it only has kinetic energy due to its motion. When the ball reaches the highest point on the ramp, it momentarily stops, meaning it has reached its maximum height and has no kinetic energy.
The total mechanical energy of the ball is given by:
πΈ = ππβ,
where π is the mass of the ball, π is the acceleration due to gravity, and β is the maximum height reached by the ball.
We need to find the initial speed π£β of the ball. The total mechanical energy can also be expressed as the sum of its translational kinetic energy and rotational kinetic energy:
πΈ = 1/2 ππ£βΒ² + 1/2 πΌπΒ²,
where πΌ is the moment of inertia of the ball and π is its angular velocity.
Since the ball is rolling smoothly, π = π£β/π, where π is the radius of the ball.
The moment of inertia of a solid ball can be represented as πΌ = 2/5 ππΒ².
Replacing the values into the equation for energy conservation, we have:
ππβ = 1/2 ππ£βΒ² + 1/2 (2/5 ππΒ²) (π£β/π)Β².
The mass π and the radius π of the ball appear in both terms of the equation, so we can simplify it to:
πβ = 1/2 π£βΒ² + 1/5 π£βΒ².
Combining the terms, we have:
πβ = 7/10 π£βΒ².
Solving for π£β, we get:
π£β = β(πβ Γ 10/7).
Now, we can substitute the given values into the equation to calculate π£β. Let's assume π = 9.8 m/sΒ² and β = 1.65 m.
π£β = β(9.8 Γ 1.65 Γ 10/7).
Calculating this expression, we find that the initial speed of the ball is approximately 6.85 m/s.