Finding the max Force ? ?

Question

ok here is the question,

An electric razor (older Braun model) has a razorhead (m = 10 g) vibrating 1.5 mm to both sides of its equlibrium position with the frequency (f = 60 Hz) of the alternating current.

Find the maximum force exerted on this razorhead.

please help me to find out this problem !!

Answer 1

The razorhead is executing Simple Harmonic motion.

Let the equation be y= A sin(2*pi* f *t + theta).

You know that A is 1.55mm

at t=0 y=0 (equilibrium poisition means no net force so y=0)

therefore theta also equals 0.

Now y= A sin( 2* Pi * f * t)

acceleration is twice differentiation of y with respect to t.

that would be a= - A (2* Pi* f)^2 Sin(2*Pi*f*t)

Since maximum value of Sin is 1

Therefore |F(max)|= mA (2*Pi*f)^2 Newton

Answer 2

To find the force on the razor head we will use the equation:

F=m*a

We have the mass, so we need to determine the acceleration the head undergoes.  The head accelerates from rest to the equilibrium point, decelerates to the end of its stroke and returns again, 60times a second. It accelerates from rest to a distance of 1.5mm (0.0015m) 4 times for each cycle. Therefore it accelerates from rest to a distance if 1.5mm in 1/240th of a second, which is 0.0042s

Calculate the acceleration:

s = ut +½*a*t²

0.0015 = 0+½*a* (0.0042)²

a = 0.0015/ 8.82*10^-6

a = 170m/s²

Now we have the data, calculate F

F=m*a

F = 0.010 *170

Force = 1.7N

Note: this would actually be the resultant force on the head. Other forces such as friction are not included. F in the equation is the resultant or net force, not the total force.