9 Questions
What does Hooke's Law describe?
The stiffness of a spring
In Hooke's Law, what does a large spring constant (k) indicate?
A stiff spring
What is the displacement (x) in Hooke's Law?
The displacement from the equilibrium position
What type of motion is simple harmonic motion?
Uniform circular motion
What does the negative sign in Hooke's Law indicate?
Force opposite to displacement
In the context of a vertical spring, what type of motion does an object exhibit?
Simple harmonic motion
What is the measure of the stiffness of a spring?
Spring constant (k)
Which factor indicates a soft spring according to Hooke's Law?
Displacement from equilibrium position
What does a small spring constant (k) suggest about a spring?
Stiffness
Study Notes
Motion of a Pendulum
- A pendulum's motion can be represented by a sinusoidal pattern when a pen is tracing its motion on a sheet of paper moving horizontally with constant speed.
- The force of a pendulum can be calculated using the formula: Ft = -mg sin θ (where Ft is the force, m is the mass, g is the acceleration due to gravity, and θ is the angle of displacement).
- For small angles, sin θ ≈ θ, which simplifies the force formula to Ft ≈ -mgθ.
Simple Harmonic Motion
- Simple harmonic motion is a type of periodic motion that follows the equation of motion: x(t) = A cos(ωt) (where x is the displacement, A is the amplitude, ω is the angular frequency, and t is time).
- The angular frequency ω is related to the frequency f by the equation ω = 2πf.
- The period T of the motion is the time taken for one complete oscillation, and is related to the frequency by the equation f = 1/T.
Physical Pendulum
- A physical pendulum has a moment of inertia I and oscillates with a period given by the formula T = 2π √(I/mgL) (where m is the mass, g is the acceleration due to gravity, and L is the length of the pendulum).
Comparing Simple Harmonic Motion with Uniform Circular Motion
- Simple harmonic motion and uniform circular motion are related, as the velocity v of an object in circular motion can be represented by the equation v = ±√(A² - x²) (where A is the amplitude and x is the displacement from the equilibrium position).
- The velocity v can also be represented by the equation v = v₀ sin θ, where v₀ is the maximum velocity and θ is the angle of displacement.
Period, Frequency, and Angular Frequency
- The period T, frequency f, and angular frequency ω are related by the equations f = 1/T and ω = 2πf.
Hooke's Law
- Hooke's Law states that the force Fs exerted by a spring is proportional to the displacement x from the equilibrium position, with the equation Fs = -kx (where k is the spring constant).
- The negative sign indicates that the force is always directed opposite to the displacement.
Damped Oscillations and Waves
- Damped oscillations occur when an oscillating system loses energy due to friction.
- Waves can be classified into different types, including mechanical waves and electromagnetic waves.
- The speed of waves on strings is dependent on the tension and linear mass density of the string.
Test your understanding of the motion of a pendulum and how it relates to vertical vibrations and horizontal movements. Explore the sinusoidal pattern traced by a pen attached to a vibrating pendulum.
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