To find the work done by the force of gravity, we can use the formula W = mgh, where m is the mass of the block, g is the acceleration due to gravity (9.8 m/s^2), and h is the vertical height of the incline. In this case, the height of the incline is given by h = 2.4sin30 = 1.2 m. Thus, the work done by the force of gravity is:
W = (5 kg)(9.8 m/s^2)(1.2 m) = 58.8 J
For the change in mechanical energy due to friction, we need to consider the work done by the frictional force. We can use the same formula W = fdcosx, where f is the force of friction and x is the angle between the force of friction and the direction of motion. In this case, the angle x is 30 degrees. Now, we need to find the magnitude of the frictional force. We can use the formula Ff = μN, where μ is the coefficient of friction and N is the normal force. The normal force is equal to the weight of the block, which is mg = (5 kg)(9.8 m/s^2) = 49 N. The coefficient of friction is not given, so we cannot calculate the exact value of the frictional force. However, we can say that the frictional force is less than or equal to μN, so we can use a range of values for μ to calculate a range of values for the change in mechanical energy. For example, if we assume μ = 0.1, we get:
Ff = (0.1)(49 N) = 4.9 N
So, the range of values for the change in mechanical energy is:
W = (4.9 N)(2.4 m)(cos30) = 11.76 J to (49 N)(2.4 m)(cos30) = 117.6 J
Therefore, the change in mechanical energy due to friction is between 11.76 J and 117.6 J. It is important to note that the actual value of the change in mechanical energy will depend on the specific value of the coefficient of friction, which is not given in this problem.