a particle vibrates according to the equation x= 20cos(16t) where x is in centimeters. determine the amplitude and frequency of the oscillation and also the position when the kinetic energy is at a maximum.
a. A=20 cm, f=5.1 Hz, x=20 cm
b. a= 20 cm, f= 2.5 Hz, x= 20 cm
c. A=20 cm, f= 2.5 Hz, x= 0 cm
d. A= 20 cm, f= 16 Hz , x=0 cm
e. the correct information is not listed in the answer selection.
I choose b. Is that correct.
To determine the amplitude and frequency of the oscillation and the position when the kinetic energy is at a maximum, we need to understand the given equation x = 20cos(16t).
1. Amplitude (A):
The amplitude of the oscillation is the maximum displacement from the equilibrium position. In this case, the coefficient of the cosine function, 20, represents the amplitude. Therefore, the amplitude is 20 cm.
2. Frequency (f):
The frequency of oscillation is the number of complete cycles per unit time. In this equation, the angular frequency is given by the value inside the cosine function, which is 16. The relationship between angular frequency and frequency is f = w/2π, where w represents angular frequency. So, the frequency is f = 16 / (2π) ≈ 16 / 6.28 ≈ 2.54 Hz, which can be approximated as 2.5 Hz.
3. Position when kinetic energy is at a maximum:
The kinetic energy is at its maximum when the velocity is at its maximum, and the velocity is at its maximum when the particle passes through the equilibrium point. In this equation, the equilibrium point occurs when x = 0 cm, which means the particle is at its midpoint in the oscillation.
Looking at the answer choices, option b. a=20 cm, f=2.5 Hz, x=20 cm does not align with our calculations. Therefore, the correct answer is e. The correct information is not listed in the answer selection.
A = 20 cm, given
when is 16 t = 2 pi ?
T = 2 *3.14159 / 16 = 0.3927 seconds period
f = 1/T = 2.55 Hz
velocity, v = -16*20 * sin 16 t
when is it max absolute value?
when 16 t = pi/2
t = 0.0982
x = 20 cos 16 * pi/32 = 20 cos pi/2 (which we should have seen before bothering to calculate v)
x= 20
so I pick B
A 1500 kg truck traveling at 80 km/h collides with another car of mass 1000 kg traveling at 30 km/h in the opposite direction. The two cars stick together after the collision. The speed immediately after the collision is
Now we have to worry about momentum, same before and after.
before:
1500 kg * 80 km/h - 1000 * 30 = 120,000 - 30,000 = 90,000 kg km/h
same after so
90,000 = (1500 +1000) v
v = 90,000 / 2500 = 36 km/h
Note - did not bother to convert km/h to m/s because the conversion cancels out
it is 1000 m / 3600 s = 1 km/ h for future reference