Answer to Question #164047 in Mechanics | Relativity for Emilia

Question

Sand from a stationary hopper falls on a moving conveyor belt at the rate of 7.64 kg/s, as shown in the figure. The belt is supported by frictionless rollers and moves at 0.466 m/s under the action of a horizontal external force supplied by the motor that drives the belt.

a) Find the sand’s rate of change of momentum in the horizontal direction. Answer in units of N.

b) Find the frictional force exerted by the belt on the sand. Answer in units of N. 

c) Find the magnitude of the external force supplied by the motor. Answer in units of N

d) Find the work done by the external force each second. Answer in units of J.

e) Find the kinetic energy acquired by the falling sand each second due to the change in its horizontal motion. Answer in units of J

Answer 1

$\frac{dm}{dt}=7.64kg/s$ $,v=0.466m/s$

a) rate of change of momentum is in horizontal

$\frac{dp}{dt}=\frac{dm}{dt}\times v=7.64\times0.466=3.56N$

b) $F_f=\frac{dp}{dt}=3.56N$

c) The friction on sand is in forward direction so its reaction on the belt is in negative direction and to keep the belt moving with the same velocity,

$F_{ext}=\frac{dp}{dt}=3.56$

d) $t=1sec$

work done, $W=Fvt=3.56\times0.466\times1=1.6590J$

e) $K.E=1/2mv^2$

$m=dm/dt\times t$

$K.E =1/2\times dm/dt\times t\times v^2=1/2 \times 7.64\times 1\times 0.466^2=0.8295$