Question:

A simple pendulum consists of a mass of 0.2 kg attached to a string of length 1.5 m. The pendulum is released from rest with an angle of 30 degrees. The motion of the pendulum can be approximated as simple harmonic motion.

1. Calculate the period of the pendulum.
2. Determine the maximum speed and maximum acceleration of the pendulum.

Answer:

1. To calculate the period of the pendulum, we can use the formula:

   

   where:

   • T is the period of the pendulum,
   • L is the length of the pendulum,
   • g is the acceleration due to gravity.

   Given that L = 1.5 m and g = 9.8 m/s², we can substitute these values into the formula:

   

   Calculating further:

   

   Therefore, the period of the pendulum is:

   (rounded to one decimal place)

2. To determine the maximum speed and maximum acceleration of the pendulum, we can use the formulas:

   • Maximum speed formula:

   • Maximum acceleration formula:

   where:

   • v is the maximum speed,
   • a is the maximum acceleration,
   • A is the amplitude of the motion,
   • is the angular frequency, given by .

   Using the period calculated earlier (T ≈ 3.1 s), we can find :

   

   Substituting the angle of 30 degrees, we find the amplitude using trigonometry:

   

   

   Finally, substitute the obtained values into the formulas:

   • Maximum speed:

   • Maximum acceleration:

   Evaluating these expressions will give the following results:

   • Maximum speed: (rounded to three decimal places)

   • Maximum acceleration: (rounded to three decimal places)

Explanation:

1. To find the period of the pendulum, we use the formula T = 2π√(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. Plugging in the given values L = 1.5 m and g = 9.8 m/s², we simplify and calculate the value of T.

2. To determine the maximum speed and maximum acceleration of the pendulum, we use the formulas v = A⋅ω and a = A⋅ω². First, we calculate the angular frequency ω by using the formula ω = 2π/T, where T is the period of the pendulum. With the calculated value of T, we substitute it into the formula and evaluate ω. Next, we find the amplitude A using the length of the pendulum and the given angle. Finally, we substitute the values of A and ω into the formulas for v and a, respectively, to obtain the maximum speed and maximum acceleration of the pendulum.