FIGURE P15.62 is a top view of an object of mass connected between two stretched rubber bands of length . The object rests on a frictionless surface. At equilibrium, the tension in each rubber band is . Find an expression for the frequency of oscillations perpendicular to the rubber bands. Assume the amplitude is sufficiently small that the magnitude of the tension in the rubber bands is essentially unchanged as the mass oscillates.
An expression for the frequency of oscillations perpendicular to the rubber bands is
1. Newton's Second Law states that the net force acting on a mass body is proportional to the acceleration of the body
2. For a particle in simple harmonic motion, the usual equation of motion is:
Where is the motion's angular frequency.
3. The angular frequency of a particle is related to its oscillation frequency as follows:
1. The mass of the object is: .
2. The length of each rubber band is: .
3. The tension in each rubber band at equilibrium is: .
The figure depicts a free-body diagram for the block as it movesvertically, with denoting the rubber bands' tension force.
When we use Newton's second law in the vertical direction from Equation (1), we get:
The sine of the angle can be calculated using the geometry shown in Figure:
since the amplitude is supposed to be modest. As a result, we can ignore in the denominator.
Substitute for into Equation (4):
Comparing Equations (2) and (5), we obtain:
Equation (3) is then used to calculate the frequency of oscillations perpendicular to the rubber bands:
The analysis of a simple pendulum assumed that the mass was a particle, with no size. A realistic pendulum is a small, uniform sphere of mass and radius at the end of a massless string, withbeing the distance from the pivot to the center of the sphere.
a. Find an expression for the period of this pendulum.
b. Suppose, typical values for a real pendulum. What is the ratio, whereis your expression from part a and is the expression derived in this chapter?
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