A spring is standing upright on a table with its bottom end fastened to the table. A block is dropped from a height 3.0 cm above the top of the spring. The block sticks to the top end of the spring and then oscillates with an amplitude of 10 cm. What is the oscillation frequency?
The oscillation frequency is f = 1.8 Hz.
1- Energy conservation principle: The sum of a system's beginning energies plus the work done on it by external forces equals the sum of the system's end energies:
Ei +W = Ef
2. Kinetic energy: An object's kinetic energy is:
when m is the organism's mass and v speed
3- Gravitational potential energy: An object's gravitational potential energy is defined as:
PE = mgh
where m, is the object's mass, and h is height.
4- Elastic Potential Energy: The elastic potential energy of a spring-like item with a spring constant k stretched or compressed x from its undisturbed position is:
5- Particle moving in a straight line with constant acceleration: If a particle moves in a horizontal path with constant acceleration , its motion is described by the kinematics equation, from which the following equation is derived:
6- The quadratic formula gives the answer to a quadratic equation of the form.
7- In simple harmonic motion, the rate of an oscillator is given by
It is important to note which does not depend on the amplitude, but rather on the mass and force constant .
The oscillation amplitude of the unit is: .
Info that is required
The oscillation frequency of the block must be determined.
Assign the block, the spring, and the earth to the system. Let the system's initial state be when the block first contacts the spring, and the system's ultimate state be when the block reaches its lowest position and the spring is entirely compressed. Assume that the system's equilibrium position is the zero-gravitational-energy configuration. Between the described starting and end states, apply the principle of energy conservation from Equation(1) to the system comprising the block, the spring, and the ground.
where as determined by Equation (2), v being the speed of the block after falling a distance h, as determined by Equation (3), y being the spacing compressed by the spring, because the season is not compressed, because there are no external pressures on the system, because the block stops temporarily at the bottom rank, as determined by Equation (3), and , is the distance squeezed by the force from the optimum configuration, as determined by Equation (4).
Let the initial state be when the block is initially lowered from rest, and the final state be when the block hits the spring for the first time. Model the block as a particle moving vertically with continuous acceleration (gravitational acceleration) and solve equation() to obtain the block's final speed, with the positive direction being downward.
The gravitational and spring forces on the block are equal at equilibrium; the gravitational force is equal to the spring pressure on the block. So
v and k from equations (9) and (10) should be substituted into equation (6): we get
Both sides are divided by :
For and , find the solution using the quadratic formula from equation ().
Because a negative sign for distance is unphysical. we used the positive root. Replace the numerical values with their equivalents.
The frequency of the block's oscillation is then calculated using equation ) ,
Substitute in the equation ()
Substitute numerical value:
An ultrasonic transducer, of the type used in medical ultrasound imaging, is a very thin disk driven back and forth in at by an electromagnetic coil.
a. The maximum restoring force that can be applied to the disk without breaking it is . What is the maximum oscillation amplitude that won't rupture the disk?
b. What is the disk's maximum speed at this amplitude?
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