A surfboard of mass S rests on:a frictionless surface. The surfboard is attached to a spring (constant K) which is in turn attached to an immobile wall. The surfboard is displaced a distance of +C from equilibrium and then is released +C +X 1.) a.) (1pts) Write the equation of motion (x as a function of t) for the surfboard (in terms of t, C, K, and S) b.) (1pts)What is the maximum velocity of the surfboardin terms of K, C, & S)? c.) (1pts) What is the maximum acceleration of the surfboard(again in terms of K, C, and S)? d.) (3pts) If a simple pendulum had the same period of oscillation, what would the length of the string be(in terms of g, S and K)? e.) (4pts) If I wish to make a physical pendulum from a uniform rod of length L and mass M that was free to swing from one end and was meant to keep the same period of oscillation as the surfboard, what must the length of the rod be? Your answer should only be in terms of g, S and K.
Solution
part a:
it will exhibit SHM with amplitude C and angular frequency w.
where w=sqrt(spring constant / mass)=sqrt(K/S)
equation is given by:
x(t)=C*cos(w*t)
=C*cos(sqrt(K/S)*t)
part b:
velocity=v=dx/dt=-C*w*sin(w*t)
so maximum velocity=C*w
=C*sqrt(K/S)
part c:
acceleration=dv/dt=-C*w^2*cos(w*t)
maximum acceleration=C*w^2
=C*(K/S)
part d:
angular frequency for pendulum is same as the spring mass system
so sqrt(g/length)=sqrt(K/S)
==>g/length=K/S
==>length=g*S/K
part e:
for a physical pendulum,
angular frequency=sqrt(mass*g*Lcm/Isupport)
where Lcm=distance of support from the center of mass of the system
Isupport=moment of inertia about the support
for the uniform rod, Isupport=mass*length^2/3
=M*L^2/3
distance of center of mass from the support=L/2
then angular frequency=sqrt(M*g*L/2/(M*L^2/3))
=sqrt(3*g/(2*L))
equating with the angular frequency of the spring,
sqrt(3*g/(2*L))=sqrt(K/S)
==>3*g/(2*L)=K/S
==>L=3*g*S/(2*K)
.
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A surfboard of mass S rests on-a frictionless surface- The surfboard i.docx
1. A surfboard of mass S rests on:a frictionless surface. The surfboard is attached to a spring
(constant K) which is in turn attached to an immobile wall. The surfboard is displaced a distance
of +C from equilibrium and then is released +C +X 1.) a.) (1pts) Write the equation of motion (x
as a function of t) for the surfboard (in terms of t, C, K, and S) b.) (1pts)What is the maximum
velocity of the surfboardin terms of K, C, & S)? c.) (1pts) What is the maximum acceleration of
the surfboard(again in terms of K, C, and S)? d.) (3pts) If a simple pendulum had the same
period of oscillation, what would the length of the string be(in terms of g, S and K)? e.) (4pts) If
I wish to make a physical pendulum from a uniform rod of length L and mass M that was free to
swing from one end and was meant to keep the same period of oscillation as the surfboard, what
must the length of the rod be? Your answer should only be in terms of g, S and K.
Solution
part a:
it will exhibit SHM with amplitude C and angular frequency w.
where w=sqrt(spring constant / mass)=sqrt(K/S)
equation is given by:
x(t)=C*cos(w*t)
=C*cos(sqrt(K/S)*t)
part b:
velocity=v=dx/dt=-C*w*sin(w*t)
so maximum velocity=C*w
=C*sqrt(K/S)
part c:
acceleration=dv/dt=-C*w^2*cos(w*t)
2. maximum acceleration=C*w^2
=C*(K/S)
part d:
angular frequency for pendulum is same as the spring mass system
so sqrt(g/length)=sqrt(K/S)
==>g/length=K/S
==>length=g*S/K
part e:
for a physical pendulum,
angular frequency=sqrt(mass*g*Lcm/Isupport)
where Lcm=distance of support from the center of mass of the system
Isupport=moment of inertia about the support
for the uniform rod, Isupport=mass*length^2/3
=M*L^2/3
distance of center of mass from the support=L/2
then angular frequency=sqrt(M*g*L/2/(M*L^2/3))
=sqrt(3*g/(2*L))
equating with the angular frequency of the spring,
sqrt(3*g/(2*L))=sqrt(K/S)
==>3*g/(2*L)=K/S
==>L=3*g*S/(2*K)
