How much work would it take to horizontally accelerate an object with a mass of #8 kg# to #3 m/s# on a surface with a kinetic friction coefficient of #1 #?

Question

How much work would it take to horizontally accelerate an object with a mass of #8    kg# to #3    m/s# on a surface with a kinetic friction coefficient of #1   #?

Nathan Ashcroft · Accepted Answer

#36J#

Concept 1 In the problem, it is important to note that there is no friction. In the formula,

#F_f=muF_N#

#mu# represents the constant of proportionality. But since it equals to #1#, #F_f=F_N#.

Concept 2 To solve the problem, you must use two formulas:

#color(blue)(|bar(ul(color(white)(a/a)W=DeltaE_kcolor(white)(a/a)|)))# where: #W=#work (joules) #DeltaE_k=#change in kinetic energy (joules)

#color(blue)(|bar(ul(color(white)(a/a)E_k=1/2mv^2color(white)(a/a)|)))# where: #m=#mass (metres) #v=#velocity (metres per second)

Solving for Work Start by breaking down #DeltaE_k# into final kinetic energy minus the initial kinetic energy.

#W=DeltaE_k#

#W=E_(k,"final")-E_(k,"initial")#

#W=1/2mv_f^2-1/2mv_i^2#

#W=1/2m(v_f-v_i)^2#

Substitute your values.

#W=1/2(8kg)(3m/s-0m/s)^2#

Solve.

#W=1/2(8kg)(3m/s)^2#

#W=color(green)(|bar(ul(color(white)(a/a)color(black)(36J)color(white)(a/a)|)))#

Carson Alexander · Answer

undefined To calculate the work done to horizontally accelerate an object to a certain velocity against friction, we can use the work-energy principle.

The work done is equal to the change in kinetic energy of the object:

$$ \text{Work} = \Delta KE $$

$$ \text{Work} = \frac{1}{2} m (v_f^2 - v_i^2) $$

Where:
- $ m $ is the mass of the object (8 kg),
- $ v_f $ is the final velocity (3 m/s),
- $ v_i $ is the initial velocity (assumed to be 0 m/s since we're starting from rest).

Plugging in the values:

$$ \text{Work} = \frac{1}{2} \times 8 \times (3^2 - 0^2) $$

$$ \text{Work} = \frac{1}{2} \times 8 \times (9 - 0) $$

$$ \text{Work} = \frac{1}{2} \times 8 \times 9 $$

$$ \text{Work} = 36 \text{ J} $$

So, it would take 36 Joules of work to horizontally accelerate the object to a velocity of 3 m/s against a surface with a kinetic friction coefficient of 1.