In an electric shaver, the blade moves back and forth over a distance of 2.2 mm in simple harmonic motion, with frequency 116 Hz. Find (a) the amplitude (in mm), (b) the maximum blade speed, and (c) the magnitude of the maximum blade acceleration.
I got that (a) is 1.1 mm, but am confused on the rest.
To solve for (b) and (c), we need to use the formula for simple harmonic motion:
1. Amplitude: We already have the correct answer for this. The amplitude is indeed 1.1 mm.
2. Maximum blade speed: The maximum blade speed occurs when the blade passes through the equilibrium position (the midpoint of its motion). At this point, the velocity of the blade is at its maximum. The maximum velocity can be calculated using the formula:
v_max = 2πfA
where f is the frequency, A is the amplitude, and π is a constant approximately equal to 3.14.
Given:
f = 116 Hz
A = 1.1 mm (or 0.0011 m, since we need to express the amplitude in meters for consistency with the SI unit system)
Plugging these values into the formula, we get:
v_max = 2π(116)(0.0011) ≈ 0.808 m/s
Thus, the maximum blade speed is approximately 0.808 m/s.
3. Magnitude of the maximum blade acceleration: The maximum acceleration occurs when the blade passes through the equilibrium position. This can be calculated using the formula:
a_max = 4π²f²A
Given the same values:
a_max = 4π²(116)²(0.0011) ≈ 33.167 m/s²
Therefore, the magnitude of the maximum blade acceleration is approximately 33.167 m/s².
To find the amplitude, (a), we can use the formula:
Amplitude = Distance / 2
Given that the blade moves back and forth over a distance of 2.2 mm, we can substitute it into the formula:
Amplitude = 2.2 mm / 2 = 1.1 mm
So you were correct in finding that the amplitude is 1.1 mm.
Moving on to finding the maximum blade speed, we can use the formula:
Maximum speed = 2 * π * frequency * amplitude
Given that the frequency is 116 Hz and the amplitude is 1.1 mm, we can substitute these values into the formula:
Maximum speed = 2 * π * 116 Hz * 1.1 mm
To simplify the calculation, we can convert Hz to s^-1 and mm to m:
Maximum speed = 2 * π * (116 s^-1) * (1.1 / 1000 m)
Calculating this, we get:
Maximum speed ≈ 0.805 m/s
Therefore, the maximum blade speed is approximately 0.805 m/s.
Lastly, to find the magnitude of the maximum blade acceleration, we can use the formula:
Maximum acceleration = (2 * π * frequency)^2 * amplitude
Given that the frequency is 116 Hz and the amplitude is 1.1 mm, we can substitute these values into the formula:
Maximum acceleration = (2 * π * 116 Hz)^2 * 1.1 mm
To simplify the calculation, we convert Hz to s^-1 and mm to m:
Maximum acceleration = (2 * π * (116 s^-1))^2 * (1.1 / 1000 m)
Calculating this, we get:
Maximum acceleration ≈ 953 m/s^2
Therefore, the magnitude of the maximum blade acceleration is approximately 953 m/s^2.