The flat-bed trailer carries two 1500-kg beams with the upper beam secured by a cable. The coefficients of static friction between the two beams and between the lower beam and the bed of the trailer are 0.25 and 0.30, respectively. Knowing that the load does not shift, determine(a) the maximum acceleration of the trailer and the corresponding tension in the cable, (b) the maximum deceleration of the trailer.

Answer :

Answer:

a) a= 8.33 m/s²,    T = 12.495 N , b)    a = 2.45 m / s²

Explanation:

a) this is an exercise of Newton's second law. As the upper load is secured by a cable, it cannot be moved, so the lower load is determined by the maximum acceleration.

We apply Newton's second law to the lower charge

            fr₁ + fr₂ = ma

The equation for the force of friction is

          fr = μ N

Y Axis

         N - W₁ –W₂ = 0

         N = W₁ + W₂

         N = (m₁ + m₂) g

Since the beams are the same, it has the same mass

        N = 2 m g

We replace

           μ₁ 2mg + μ₂ mg = m a

          a = (2μ₁ + μ₂) g

          a = (2 0.30 + 0.25) 9.8

          a= 8.33 m/s²

Let's look for cable tension with beam 2

          T = m₂ a

          T = 1500 8.33

          T = 12.495 N

b) For maximum deceleration the cable loses tension (T = 0 N), so as this beam has less friction is the one that will move first, we are assuming that the rope is horizontal

           fr = m₂ a₂

           N- w₂ = 0

          N = W₂ = mg

          μ₂ mg = m a₂

          a = μ₂ g

          a = 0.25 9.8

          a = 2.45 m / s²

a) The maximum acceleration of the trailer and the corresponding tension in the cable is [tex]8.33 m/s^2[/tex] and 12.495 N

b)  The maximum deceleration of the trailer is [tex]2.45 m / s^2[/tex]

  • The calculation is as follows;

Here we  apply Newton's second law to the lower charge

[tex]fr_1 + fr_2 = ma[/tex]

The equation for the force of friction is

[tex]fr = \mu N[/tex]

Y Axis

[tex]N - W_1 -W_2 = 0\\\\N = W_1 + W_2\\\\N = (m_1 + m_2) g[/tex]

Because the beams are the same, it has the similar mass

N = 2 m g

Now

[tex]\mu_1 2mg + \mu_2 mg = m a\\\\a = (2\mu_1 + \mu_2) g\\\\a = (2 0.30 + 0.25) 9.8\\\\ a= 8.33 m/s^2[/tex]

Now  cable tension with beam 2

[tex]T = m_2 a[/tex]

T = 1500 8.33

T = 12.495 N

b)

the maximum deceleration of the trailer is

[tex]fr = m_1 a_2\\\\N- w_2 = 0\\\\N = W_2 = mg\\\\ \mu_2 mg = m a_2\\\\a = \mu_2 g\\\\a = 0.25 (9.)8\\\\ a = 2.45 m / s^2\\\\[/tex]

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