Simple Pendulum Period and Gravity

Simple Pendulum

• The period of a simple pendulum is determined solely by its length and the acceleration due to gravity.
• Mass does not influence the pendulum's period.
• The formula for the period (T) is given by:
  ( T = 2\pi \sqrt{\frac{L}{g}} )
  where L is the length and g is the acceleration due to gravity.

Example Calculations

• For a pendulum with a length of 50 cm (0.5 m) and g = 9.8 m/s²:
  ( T = 2\pi \sqrt{\frac{0.5}{9.8}} \approx 1.42 ) seconds.

• For a pendulum of 100 cm (1 m) and a period of 0.65 seconds, the acceleration due to gravity can be calculated using:
  ( g = \frac{4\pi^2 L}{T^2} ), resulting in g ≈ 9.47 m/s².

Activity to Measure Gravity

• Use a 1 m string attached to a metal ball to form a simple pendulum.
• Measure the period for 10 complete oscillations using a stopwatch to calculate local gravitational acceleration.
• Consider how measurement accuracy can be improved.

Spring Mechanics

• A mass attached to a spring oscillates about its equilibrium position (x = 0), where the spring is neither stretched nor compressed.

• Displacements from equilibrium lead to periodic motion, described by Hooke's Law:
  ( F_{\text{res}} = -kx ), where k is the spring constant.

• The restoring force (F_res) is directly proportional to the displacement (x) from the equilibrium position.

• A stiffer spring has a larger spring constant (k), resulting in stronger restoring forces for smaller displacements.

• The actual value of k can be determined using:
  ( k = \frac{F_{\text{app}}}{x} ), with units of N/m.

Oscillation Dynamics

• The motion of an oscillating object involves continuous changes in displacement as it moves back and forth across the equilibrium position.
• In one complete cycle, the object's displacement decreases as it approaches equilibrium and then increases in the opposite direction before reversing.

Hooke's Law Demonstration

• Hooke's Law illustrates the relationship between force exerted by a spring and its displacement.
• During demonstrations, different types of springs can be compared based on stiffness, illustrating practical applications of spring mechanics.

Calculate the period of a simple pendulum with a given length, and understand how it's affected by the acceleration due to gravity. Learn how to use pendulums to measure gravity. Solve a problem to find the period of a pendulum with a length of 50 cm.