A 10kg particle undergoes simple harmonic motion with an amplitude of 2.0mm, a maximum acceleration of 8.0x10^3 m/s^2, and an unknown phase constant (phi) What are:
a.) the period of the motion
b.) the maximum speed of the particle
c.) total mechanical energy of the oscillator
What is the magnitude of the force on the particle when the particle is at:
d.) its maximum displacement
e.) half its maximum displacement
I have the answers from the back of the book, but I don't know which equations to use! Any help would be very appreciated!
I am going to assume you are in calculus.
x(t)=2E-4 * sin(wt+Phi)
v(t)=w*2E-4 * cos(wt+Phi)
a(t)=w^2*2E-4 sin(wt+Phi)
You know the w=2PI/period, so given max a, find w, and then period.
then max speed.
total ME? 1/2 m (maxv)^2
for the last q, remember F=ma
You know max acceleration occures when the sin term is 1, so max a= w^2*2E-4
i rekomend not pgysics as coorse ligjht
To find the answers to these questions, we can use the equations of simple harmonic motion. Let's go through each question step by step:
a.) The period of simple harmonic motion can be found using the formula T = 2π/ω, where T is the period and ω is the angular frequency. The angular frequency can be found using the formula ω = 2πf, where f is the frequency. Now, the maximum acceleration (a_max) can be related to the angular frequency using the formula a_max = ω^2A, where A is the amplitude. From this equation, you can solve for ω using the given values of a_max and A. Once you have ω, you can find the period using the formula T = 2π/ω.
b.) The maximum speed of the particle can be found using the formula v_max = ωA, where ω is the angular frequency and A is the amplitude. Once you have found the value of ω from part (a), you can substitute it into this equation along with the given value of A to find the maximum speed.
c.) The total mechanical energy (E) of the oscillator can be found using the equation E = (1/2)kA^2, where k is the spring constant and A is the amplitude. However, in this case, the spring constant is not given directly. Instead, we can relate it to the angular frequency using the formula k = mω^2, where m is the mass of the particle. Now, you can substitute the value of ω obtained from part (a), the given value of A, and the given value of mass to find the total mechanical energy.
d.) The magnitude of the force on the particle when it is at its maximum displacement can be found using the formula F = ma, where m is the mass of the particle and a is the acceleration. The magnitude of the force will be equal to the maximum acceleration (a_max) given in the question.
e.) The magnitude of the force on the particle when it is at half its maximum displacement can be found using the formula F = -kx, where k is the spring constant and x is the displacement from the equilibrium position. In this case, the displacement will be half the amplitude (A/2). The spring constant can be related to the angular frequency using the formula k = mω^2, where m is the mass of the particle. Substitute the given values of m and A/2 into this equation and calculate the magnitude of the force.
Using these equations and the given values, you should be able to find the answers to each question.