Find the acceleration and force of a 20 kg box pushed by 80 kg person.

Question

A person weighing 80 kg applies a horizontal force to a 20 kg box to set it in motion. We give μs = 0.8 for the person and μk = 0.4 for the box

a) Assuming the person remains stationary, what is the modulus of the maximum possible acceleration of the box? (Answer is 27.4 m/s^2)

b) Use the answer to the previous question to determine the modulus of the force exerted by the box on the person. (Answer is 627 N)

Answer 1

a) In order to find acceleration of the box we will apply the Newton 's 2-d law to the box. There are 4 forces acting on the box:

1. weight mg, where m = 20 kg - mass of the box
2. normal reaction force N₁
3. external force from the person F (let's assume its direction to the right horizontally)
4. f₁ - kinetic friction from the surface - direction to the left (against motion of the box)

Let's write newton's 2-d law in vector form:

mg + N₁ + F + f₁ = ma

Let's select direction of the x-axis to the right and y-axis perpendicular to it up. Then in scalar form we have:

x: F - f₁ = ma (1)

y: N₁ - mg = 0 (2)

We know that kinetic friction force f₁ = µ_kN₁ and from (2) N_{1 =} mg, then f_{1 =} µ_kmg

Substituting it in (1), we will get

F - µ_kmg = ma.

From this equation

a= ( F - µ_kmg / m (3)

Because the person is stationary, the maximum force that he can apply is equal to the static frictional force that acts on him from the surface, i.e. F = f₂.

Similar to previous case, we can find that f₂ = µ_sMg, where µ_s = 0.8 - coefficient of static friction, M = 80 kg - mass of the person, and g - acceleration due to gravity. so F = µ_sMg.

Now we finally writing:

a = g(µ_sM - µ_km) / m (4)

If in equation (4) you plug in the numbers, you will get a = 27.4 m/s²

b) To find the magnitude of the force exerted by the box on the person, we will use Newton's 3-d law. We already found that the person exerted on the box the force F = µ_sMg. Hence, the magnitude of force exerted by the box on the person has the same value. If you put in this formula the values of all variables, you will obtain 627 N.

P.S. Alara, thank you very much for the problem. I enjoyed doing it. Hope my solution will help you understand physics better. Good luck!