A spring with a spring constant of \(500 . \mathrm{N} / \mathrm{m}\) is used to propel a 0.500 -kg mass up an inclined plane. The spring is compressed \(30.0 \mathrm{~cm}\) from its equilibrium position and launches the mass from rest across a horizontal surface and onto the plane. The plane has a length of \(4.00 \mathrm{~m}\) and is inclined at \(30.0^{\circ} .\) Both the plane and the horizontal surface have a coefficient of kinetic friction with the mass of \(0.350 .\) When the spring is compressed, the mass is \(1.50 \mathrm{~m}\) from the bottom of the plane. a) What is the speed of the mass as it reaches the bottom of the plane? b) What is the speed of the mass as it reaches the top of the plane? c) What is the total work done by friction from the beginning to the end of the mass's motion?

We will first find the speed of the mass just after leaving the spring and before reaching the inclined plane. To do this, we need to remember that the total mechanical energy (sum of kinetic and potential energies) is conserved when external forces do no work. Potential energy stored in the compressed spring (Us) is: Us = (1/2) * k * x^2 Kinetic energy of the mass after leaving the spring (K1) is: K1 = (1/2) * m * v_1^2 (where v_1 is the speed of the mass at this point) Since no external force is doing work, we have: Us = K1 (1/2) * k * x^2 = (1/2) * m * v_1^2 Now, we can solve for v_1. Next, we should find the loss of kinetic energy due to the frictional force along the horizontal distance d before entering the inclined plane. Work done by friction (Wf) along the horizontal distance is: Wf = -μ * m * g * d (Negative sign because friction opposes the movement) According to the work-energy theorem, the loss of kinetic energy will equal the work done by friction. So, we can write: K1 - K2 = -Wf (where K2 is the kinetic energy just before entering the incline) Now, we can solve for K2 and find the speed of the mass (v_2) at the bottom of the inclined plane: K2 = (1/2) * m * v_2^2

To find the speed at the top of the inclined plane, we need to consider the gravitational potential energy, kinetic energy, and energy loss due to friction along the inclined plane. First, let's find the height (h) at the top of the inclined plane: h = L * sin(θ) Now we can find the gravitational potential energy at the top of the inclined plane (Ug): Ug = m * g * h Next, let's find the energy loss due to friction along the inclined plane (Wf_incline): Wf_incline = -μ * m * g * L * cos(θ) (Negative sign due to energy loss) According to the work-energy theorem, for the inclined plane portion, we have: K2 + Ug - K3 = -Wf_incline (where K3 is the kinetic energy at the top of the inclined plane) Now, we can solve for K3 and find the speed of the mass (v_3) at the top of the inclined plane: K3 = (1/2) * m * v_3^2

The total work done by friction from the beginning to the end of the mass's motion is the sum of the work done along the horizontal surface and the inclined plane: Wf_total = Wf + Wf_incline By calculating Wf_total, we'll find the total work done by friction through the entire motion of the mass.