Question

The only force F~ acting on a body of mass m (in kg) changes in time t according to F~ (t) = (F0 sin(ωt), F0 cos(ωt)) where F0 is a constant with units of N and the greek letter ω is a constant with units of radians/s. (Technically a radian is dimensionless, but I’ve explicitly stated it to clarify the problem.) Assume all arbitrary constants that appear due to indefinite integration are zero.

(a) What is the acceleration vector of the body as a function of time?

(b) What is the velocity vector of the body as a function of time?

(c) What is the displacement vector (from the origin) of the body as a function of time?

(d) Let F0, ω, and m all equal 1 (in appropriate units). Draw a coordinate system and add the initial displacement and velocity vectors to the coordinate system so that the displacement vector starts at the origin and the velocity vector starts at the end of the displacement vector. You can use an arbitrary length for the velocity vector.

(e) Using the same diagram and procedure created for part (d), add vectors corresponding to times t = π/2, π, and 3π/2 s. This time, make sure the magnitude of your velocity vectors are correctly proportioned relative to the size of the first velocity vector you drew in part (d).

(f) What kind of motion is this object experiencing?

Answer 1

force vector=F(t)=(F0*sin(w*t), F0*cos(w*t))

part a:

acceleration vector=force/mass

=F(t)/m=(F0*sin(w*t)/m, F0*cos(w*t)/m) m/s^2

part b:

if velocity is v(t),

then acceleration=dv/dt

==>dv/dt=(F0*sin(w*t)/m, F0*cos(w*t)/m)

==>dv=(F0*sin(w*t)*dt/m, F0*cos(w*t)*dt/m)

integrating both sides,

v(t)=(-(F0*cos(w*t)/(w*m)), (F0*sin(w*t)/(m*w))) m/s

part c:

if displacement is x(t),

v(t)=dx/dt

==>dx/dt=(-(F0*cos(w*t)/(w*m)), (F0*sin(w*t)/(m*w)))

==>dx=(-(F0*cos(w*t)*dt/(w*m)), (F0*sin(w*t)*dt/(m*w)))

integrating both sides,

x=(-(F0*sin(w*t)/(w^2*m)), (-F0*cos(w*t)/(m*w^2))) m

part d:

as F0=w=m=1

x(t)=-(sin(t),cos(t)) m

at t=0, x(t)=(0,-1) m

v(t)=(-cos(t),sin(t)) m/s

at t=0,

v(t)=(-1,0) m/s

vector diagram:

0.5 displacement vector 0.5- -1 :velocity vector 5 0.5 x component of the vector 1.5 -1 0 0.5