Final answer:
The velocity of the bob of a simple pendulum at its mean position M needed to rise to a vertical height of 10cm is approximately 1.4 m/s, using the conservation of energy principle and the known value of g = 9.8 m/s^2.
Step-by-step explanation:
To find the velocity of the bob of a simple pendulum at its mean position M, we can use the principle of conservation of energy. When the pendulum bob is at the highest point of its swing (10cm above the mean position), all its kinetic energy at the mean position will have been converted into potential energy. The amount of potential energy at the highest point can be calculated as PE = mgh, where m is the mass of the bob, g is the acceleration due to gravity, and h is the height it rises to. This potential energy should be equal to the kinetic energy at the mean position, which is given by KE = 0.5 * m * v^2, where v is the velocity we need to find.
In this case, the mass of the bob cancels out from both sides of the equation, leaving us with gh = 0.5 * v^2. Using the given value of g = 9.8 m/s^2 and h = 0.10 m, we can solve for v. Double the value of g * h and then take the square root to find v, resulting in v = sqrt(2 * g * h).
v = sqrt(2 * 9.8 m/s^2 * 0.10 m) which calculates to approximately 1.4 m/s, making option (a) the correct answer.