A small mass attached to a spring oscillates with simple harmonic motion with amplitude of 70mm,taking 13 seconds to make 40 complete oscillations.Calculate the following showing steps;
a.Its angular ferquency
b.Its maximum speed
c.Its maximum acceleration
d.Its speed when displacement from equilibrium is 10mm
e.Its speed when displacement from equilibrium is 20mm
ΔT = t/N = 13/40 = 0.325 s.
a.
ω = 2•π/T = 2•π/0.325 = 19.33 rad/s.
b.
v(max) = A•ω= 70•10^-3•19.33 = 1.353 m/s
c.
a(max) =A•ω^2 = 26.16 rad/s^2.
d.
x= A•cos(ω•t+α) = 10 mm
cos(ω•t+α) = x/A= 10/70 =1/7.
sin(ω•t+α) = sqrt(1 – (cos(ω•t+α))^2) = 0.9897,
v = - A• ω• sin(ω•t+α) = 70•10^-3•19.33•0.9897 = -1.34 m/s.
e. similar to "d".
a. To calculate the angular frequency (ω), we can use the formula:
ω = 2π / T
where T is the time taken for one complete oscillation.
Given that the mass takes 13 seconds to complete 40 oscillations, we can find the time for one complete oscillation:
T = 13 s / 40 oscillations
Now, we can calculate the angular frequency:
ω = 2π / T
b. The maximum speed (v_max) of the oscillating mass can be found using the formula:
v_max = ω * A
where A is the amplitude of the oscillation.
Given that the amplitude is 70 mm, and we have already calculated the angular frequency (ω) in part (a), we can substitute these values into the formula to find the maximum speed.
v_max = ω * A
c. The maximum acceleration (a_max) can be calculated using the formula:
a_max = ω^2 * A
Using the value of angular frequency (ω) obtained in part (a) and the amplitude (A) of 70 mm, we can calculate the maximum acceleration.
a_max = ω^2 * A
d. To find the speed of the oscillating mass when the displacement from equilibrium is 10 mm, we can use the formula:
v = ω * x
where x is the displacement from equilibrium.
Given a displacement of 10 mm, and the angular frequency (ω) from part (a), we can calculate the speed using this formula.
v = ω * x
e. Similarly, to find the speed of the oscillating mass when the displacement from equilibrium is 20 mm, we can use the same formula as in part (d):
v = ω * x
Given a displacement of 20 mm and the angular frequency (ω) from part (a), we can calculate the speed using this formula.
v = ω * x