Vibration and energy
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- When a family of three with a total mass of 150kg steps into their 1100kg car, the car's spring compresses 2.0cm.
- A spring is 45cm long when a weight of 65N hangs from it but is 96cm when a weight of 190 N hangs from it. What is the spring constant?
- A spring stretches 0.120m vertically when a mass of 0.400kg is attached to its end. Then spring is set up horizontally with the 0.400kg resting on a frictionless surface. The mass is pulled so that the spring is stretched 0.200m from the equilibrium position.
- An object with mass 4.0kg is attached to a spring with spring stiffness constant k = 200N/m and is performing the simple harmonic motion. When the object is 0.04m from its equilibrium position, it is moving with a speed of 0.62m/s.
- A mass sitting on a horizontal, frictionless surface is attached to one end of a spring; the other end is fixed to a wall. 2.0J of work is required to compress the spring by 0.14m.
If the mass experiences a maximum acceleration of 18 m/s2;
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Topic Notes
In this lesson, we will learn:
- Vibrating and oscillating systems
- Hooke’s law
- How to find the amount of energy stored in a vibrating system?
Notes:
- There are many examples of vibrating objects around us, an object attached to the end of spring, guitar strings, ruler held firmly at the end of the table.
- During vibration (oscillation) the object moves back and forth about a fixed position called “equilibrium position”.
- Let’s consider a mass vibrating at the end of a uniform spring.;
(b) The mass is oscillating; the restoring force tries to pull the mass back to its equilibrium position.
(c) The mass is oscillating; the restoring force tries to push the mass back to its equilibrium position.
Hooke’s Law
- The magnitude of the restoring force is directly proportional to the displacement x;
F∝x - The direction of the restoring force is always opposite to the displacement which is indicated by a minus sign in the equation.
F=−kx (Hooke’s Law)
F: Force exerted by the spring on the mass
K: Spring constant
x: Displacement
Energy
- As we know the mechanical energy of a system is the sum of kinetic and potential energies.
- In the case of the spring-mass system, the potential energy would be in the form of elastic potential energy in the spring which is calculated using the following equation;
Therefore; the mechanical energy of the system is;
(a) At the extreme points where the mass stops momentarily to change the direction; v = 0, x = A (amplitude, maximum displacement)
(b) At the equilibrium position the mass moves with maximum velocity;
From (1) and (2);
21kA2=21mvmax2⇒Vmax2=(mk)A2
(c) At intermediate points, the energy is a part kinetic and part potential;
- From the above equation we can find velocity as the function of position;
This gives the velocity of the object at any position.
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