   Chapter 13, Problem 33P

Chapter
Section
Textbook Problem

Given that x = A cos (ωt) is a sinusoidal function of time, show that v (velocity) and a (acceleration) are also sinusoidal functions of time. Hint: Use Equations 13.6 and 13.2.

To determine
The nature of velocity and acceleration, when the position is a sinusoidal function of time.

Explanation

Given info: The position is x=Acos(ωt) .

When the system is in the turning points, the total energy is stored in the form of elastic potential energy. From conservation of energy,

12kA2=12mv2+12kx2

• k is force constant
• A is the amplitude
• m is the mass of the object
• x is the displacement from equilibrium

Solving for v ,

v=±km(A2x2) (1)

The angular frequency ω is,

ω=km (2)

Substitute x=Acos(ωt) in (1), and using (2),

v=±km(

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