An object with a mass of #7 kg# is on a surface with a kinetic friction coefficient of # 8 #. How much force is necessary to accelerate the object horizontally at # 21 m/s^2#?
1 Answer
Explanation:
Newton's second law of motion states that the sum of the forces acting on an object is equal to its mass multiplied by its acceleration.
Mathematically speaking,
#color(blue)(|bar(ul(color(white)(a/a)F_"net"=macolor(white)(a/a)|)))# where:
#F_"net"=# net force
#m=# mass#(kg)#
#a=# acceleration#(m/s^2)#
In your case, we will start off by letting the up and forward directions be positive.
Step 1
List out all the forces acting on the object.
#F_N+F_g+F_(app)+F_f=ma#
Since the object is not changing in the
#color(red)cancelcolor(black)(F_N)+color(red)cancelcolor(black)(F_g)+F_(app)+F_f=ma#
So we are left with,
#F_(app)+F_f=ma#
Step 2
Since you are looking for the force necessary to accelerate the object, isolate for
#F_(app)=ma-F_f#
Step 3
In the equation,
#F_(app)=ma-mu_kF_N#
Note: Since the object is not changing vertically,
#F_(app)=ma-mu_kmg#
Factoring out
#F_(app)=m(a-mu_kg)#
Step 4
Plug in your known values.
#F_(app)=(7kg)[21m/s^2-(8)(-9.81m/s^2)]#
Solve.
#F_(app)=696.36N#
Rounding off the answer to one significant figure,
#F_(app)~~color(green)(|bar(ul(color(white)(a/a)color(black)(700N)color(white)(a/a)|)))#