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 #3 #?

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 #3  #?

Adrian Ayala · Accepted Answer

The work is #=36J#

The mass is #m=8kg#

The acceleration due to gravity is #g=9.8ms^-2#

The coefficient of kinetic friction is #mu_k=3#

The normal force is #N=mg#

The force of friction is

#F_r=mu_kN=mu_kmg#

According to Newton's Second Law

#F=ma#

The acceleration is

#a=F/m=(mu_kmg)/m=mu_kg#

The speed is #v=3ms^-1#

The initial speed is #u=0ms^-1#

Apply the equation of motion

#v^2=u^2+2as#

The distance is

#s=v^2/(2a)=9/(2*mu_kg)=9/(2*3*9.8)=0.15m#

The work done is

#W=F*s=mu_kmg*s=3*8*9.8*0.15=36J#

Isabella Alvarez · Answer

undefined To calculate the work required to horizontally accelerate an object with a mass of 8 kg to 3 m/s on a surface with a kinetic friction coefficient of 3, you can use the equation:

$$ W = \frac{1}{2} m v^2 + f_k d $$

where $W$ is the work done, $m$ is the mass of the object (8 kg), $v$ is the final velocity (3 m/s), $f_k$ is the kinetic friction coefficient (3), and $d$ is the distance over which the force is applied.

First, you need to find the distance $d$ using the equation:

$$ f_k = \mu_k N $$

where $\mu_k$ is the coefficient of kinetic friction and $N$ is the normal force. Since the object is horizontally accelerating, the normal force $N$ is equal to the weight of the object, which is $mg$, where $g$ is the acceleration due to gravity (9.8 m/s²).

$$ f_k = \mu_k mg $$

$$ d = \frac{v^2}{2\mu_k g} $$

Substitute the given values into the equation:

$$ d = \frac{(3\, \text{m/s})^2}{2(3)(9.8\, \text{m/s}^2)} $$

$$ d \approx 0.153\, \text{m} $$

Now, you can calculate the work done:

$$ W = \frac{1}{2} m v^2 + f_k d $$

$$ W = \frac{1}{2}(8\, \text{kg})(3\, \text{m/s})^2 + (3)(0.153\, \text{m}) $$

$$ W \approx 36\, \text{J} $$

So, it would take approximately 36 joules of work to horizontally accelerate the object to 3 m/s on a surface with a kinetic friction coefficient of 3.

Lily Adams · Answer

undefined To horizontally accelerate an object with a mass of 8 kg to 3 m/s on a surface with a kinetic friction coefficient of 0.3, it would take approximately 36 joules of work.