Final answer:
The correct answer is (b) mg. For a pendulum given horizontal velocity at the lowest point, the tension in the string at the highest point will be equal to the weight of the bob, which is mg. This is because all the kinetic energy is converted to potential energy, leaving no excess energy for centripetal force.
Step-by-step explanation:
The question deals with the tension in a simple pendulum at its highest point when given an initial horizontal velocity at its lowest point. When the pendulum reaches its highest point, all of its kinetic energy is converted into potential energy, and the only forces acting on the bob are the tension in the string and the gravitational force. At this point, the tension in the string must provide enough centripetal force to keep the mass moving in a circular path while also counteracting the gravitational force. Assuming the bob reaches the highest point without losing energy, the tension in the string (T) at this point would have to equal the weight of the bob (mg) plus the centripetal force required to keep it moving in a circle.
If we denote the maximum velocity that the bob can have at the top point before the string goes slack as v, then the centripetal force needed is mv^2/l. At the top, the speed will be zero if it just manages to reach there. Therefore, using energy conservation (mgh = (1/2)mv^2 at the bottom), you can find that the velocity at the bottom is v = √(2gl). However, at the top, all this kinetic energy is converted into potential energy; thus, there is no leftover for centripetal force, and the tension in the strings becomes equal to the weight of the bob (mg). Thereby, the correct answer is (b) mg.