A metal sphere falls with terminal velocity through cylinder filled with oil.If sphere experiences a viscous force of kv(k is constant), what is the kinetic energy of the sphere(express in m,g,k form)?
1 Answer
This is what I get
Explanation:
When metal sphere is moving at terminal velocity in a cylinder filled with oil, the forces acting on it are in equilibrium.
#"Frictional force" uarr="weight" darr-"upthrust" uarr# .....(1)
Given frictional force
Hence above equation becomes
#kv_0=(m_s-m_o)g#
where#v_0# is the terminal velocity,#m_s# is mass of metal sphere and#m_o# is mass of oil displaced.
#=>v_0=(m_s-m_o)g/k# ......(2)
Kinetic energy of the sphere is given as
#KE=1/2m_sv_0^2#
Inserting value of terminal velocity from (2) we get
#KE=1/2m_s((m_s-m_o)g/k)^2#
#KE=1/2m_sg^2/k^2(m_s-m_o)^2#
.-.-.-.-.-.-.-.-.-.-.-
Using Stokes Law for the frictional force (of viscosity) on a small sphere moving through a viscous fluid for LHS of (1) we get
#6pietaalphav_0=4/3pialpha^3g(rho_s-rho_o)# ......(1)
where#η# is viscosity of oil,#alpha# is radius of the metal sphere,#v_0# is terminal velocity,#g# is acceleration due to gravity,#ρ_s# is density of the material of sphere and#rho_o# is density of oil.
#=>k=6pietaalpha# ,#m_s=4/3pialpha^3rho_s and m_o=4/3pialpha^3rho_o# in the given question.
