(3 marks) The square root of a real number 1 this has no solution because a square root can't be negative, $x-3$ must be greater of equal to $\sqrt{x-3}$ 5 is negate, the square root of a real number can't be negative, The period, $T$, of a pendulum can be approximated by the formula $\approx 2 \pi \sqrt{\frac{L}{g}}$, where $L$ is the length of the pendulum and $g$ is the gravitational constant. What is the approximate length of the pendulum if it has a period of 2 s ? Note: On Earth the gravitational constant is $9.8 m / s ^2$. (3 marks)
Answer (1)
Substitute the given values into the formula: 2 = 2 π 9.8 L .
Divide both sides by 2 π : π 1 = 9.8 L .
Square both sides: π 2 1 = 9.8 L .
Multiply both sides by 9.8 : L = π 2 9.8 ≈ 0.9929 . The approximate length of the pendulum is 0.9929 meters .
Explanation
Understanding the Problem We are given the formula for the period of a pendulum: T ≈ 2 π g L , where L is the length of the pendulum, and g is the gravitational constant. We are given that the period T = 2 seconds and the gravitational constant g = 9.8 s 2 m . We want to find the length L of the pendulum.
Substituting the Values We substitute the given values into the formula: 2 = 2 π 9.8 L .
Isolating the Square Root Divide both sides of the equation by 2 π : π 1 = 9.8 L .
Squaring Both Sides Square both sides of the equation: π 2 1 = 9.8 L .
Solving for L Multiply both sides of the equation by 9.8 to solve for L : L = π 2 9.8 .
Calculating the Length Now, we calculate the value of L : L ≈ 3.1415 9 2 9.8 ≈ 9.8696 9.8 ≈ 0.9929 . Therefore, the approximate length of the pendulum is 0.9929 meters.
Final Answer The approximate length of the pendulum is approximately 0.9929 meters.
Examples
Understanding the period of a pendulum has practical applications in various fields. For example, clockmakers use pendulums to regulate the timing of clocks. Civil engineers use the principles of pendulum motion to design structures that can withstand vibrations, such as bridges and skyscrapers. In sports, understanding pendulum motion can help athletes improve their performance in activities like swinging a golf club or batting a baseball. By understanding the relationship between the length of a pendulum and its period, we can design and analyze systems that rely on oscillatory motion.
