Question

Two students are conducting experiments with identical 100 g mass simple pendulums, each raising the bob 5 m above the mean position. The graph depicting the relationship between distance from the mean position and velocities is presented in the diagram.

1. Demonstrate, using mathematical calculations, that the path followed by student 1 is devoid of friction.
2. Does the situation involving student 2 exhibit a breach of the law of conservation of energy?

Answer 1

Given Data: Mass of each bob (m) = 100 g = 0.1 kg

Initial height above mean position (h) = 5 m

Acceleration due to gravity (g) = 10 m/s²

(a) For Student 1:

1. Initial Potential Energy at Extreme Position:

P.E. = m · g · h

= 0.1 × 10 × 5

= 5 J

2. Maximum Kinetic Energy at Mean Position (v = 10 m/s):

\[ \text{K.E.} = \frac{1}{2}\, mv^{2} \]

\[= \frac{1}{2} \times 0.1 \times (10)^{2} \]

= 5 J

Since Initial P.E. (5 J) = Maximum K.E. (5 J), no mechanical energy is lost. Hence, the path is devoid of friction.

(b) For Student 2:

1. Maximum Kinetic Energy at Mean Position (v = 8 m/s):

\[ \text{K.E.} = \frac{1}{2}\, mv^{2} \]

\[= \frac{1}{2} \times 0.1 \times (8)^{2} \]

= 3.2 J

2. Energy Loss = 5 J − 3.2 J

= 1.8 J

No, it does not breach the law. The missing 1.8 J of energy is simply transformed into heat and sound energy due to air resistance. Since total energy remains constant, the law of conservation of energy is perfectly obeyed.