A block with a mass of 2.00 kg is placed against a spring on a frictionless incline at an angle of 30 degrees. The block is not attached to the spring. The spring, which has a spring constant k of 19.6 N/cm, is compressed by 20 cm and then released. (a) What is the elastic potential energy of the compressed spring? (b) What is the change in gravitational potential energy of the block-Earth system as the block moves from the release point to its highest point on the incline? (c) How far along the incline is the highest point from the release point?

(a) The elastic potential energy (EPE) stored in a compressed spring can be calculated using the formula EPE = 1/2 * k * x^2, where k is the spring constant and x is the compression distance. However, we need to convert the spring constant to N/m because it is given in N/cm.

k = 19.6 N/cm × 100 cm/m = 1960 N/m

Now we can calculate the EPE with x = 20 cm = 0.20 m:

EPE = 1/2 * k * x^2 EPE = 1/2 * 1960 N/m * (0.20 m)^2 EPE = 0.5 * 1960 * 0.04 EPE = 39.2 J

So, the elastic potential energy of the compressed spring is 39.2 Joules.

(b) The change in gravitational potential energy (GPE) of the block-Earth system as the block moves from the release point to its highest point on the incline can be equated to the elastic potential energy at the release point since there are no other external forces doing work (neglecting air resistance and assuming a frictionless incline). This is due to the conservation of energy where the initial elastic potential energy will be converted into gravitational potential energy.

Therefore, the change in GPE will be the same amount as the EPE calculated above:

Change in GPE = EPE = 39.2 J

(c) To calculate how far along the incline the highest point is from the release point, we will first convert the change in GPE into the height gained by the block, and then use trigonometry to find the distance along the incline.

First, find the height (h) using the formula for gravitational potential energy GPE = mgh, where m is the mass, g is the acceleration due to gravity (9.8 m/s²), and h is the height:

Change in GPE = mgh 39.2 J = 2.00 kg * 9.8 m/s² * h h = 39.2 J / (2.00 kg * 9.8 m/s²) h = 39.2 J / 19.6 N h = 2 m

So the block rises by 2 meters along the incline. Now to find the distance along the incline, d, we use the trigonometric relation:

sin(θ) = opposite/hypotenuse sin(30°) = h/d sin(30°) = 2 m / d d = 2 m / sin(30°) d = 2 m / 0.5 d = 4 m

Thus, the highest point on the incline is 4 meters from the release point.