Final answer:
Using conservation of energy, you can calculate the linear speed of a solid ball rolling down a ramp by equating the potential energy at the top (mgh) with the kinetic energy at the bottom (translational and rotational). Solve for the linear speed v after expressing all energies in terms of v.
Step-by-step explanation:
Finding the Linear Speed of a Solid Ball Rolling Down a Ramp
To find the linear speed of a solid ball at the bottom of a ramp, we can use the principle of conservation of energy. The ball has potential energy at the top of the ramp, which is converted into translational kinetic energy and rotational kinetic energy as it rolls down. The ball starts from rest, so the initial kinetic energy is zero. Therefore, all the potential energy will be converted into kinetic energy by the time the ball reaches the bottom of the ramp.
The potential energy (PE) at the top of the ramp is given by:
Here, m is the mass of the ball (5.65 kg), g is the acceleration due to gravity (9.81 m/s2), and h is the height of the ramp. We can find h using the length of the ramp (L = 4 m) and the angle of incline (θ = 12°) with the relationship h = L sin(θ).
The final kinetic energy (KE) at the bottom of the ramp is the sum of translational kinetic energy (1/2 m v2) and rotational kinetic energy (1/2 I ω2), where I is the moment of inertia of a solid sphere (I = 2/5 m R2) and ω is the angular velocity related to the linear velocity v by the radius R (0.36 m) such that v = R ω.
By setting the initial potential energy equal to the final kinetic energy and solving for v, we can find the linear speed at the bottom of the ramp. Here are the steps:
- Calculate the height h using the sine function.
- Compute the initial potential energy PE.
- Write the equation for conservation of energy: PE = 1/2 m v2 + 1/2 (2/5 m R2) (v/R)2.
- Solve for v.
After calculations, you will get the linear speed v at the bottom of the ramp.