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下午9:08 9月24日周二

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Back HW3-NL.pdf 5) A highway exit ramp is being designed...

Sep 25, 2024

Solution by Steps

step 1

First, we need to find the net force acting on the box. The box is pushed by two forces,

F_1

and

F_2

. We can resolve these forces into their x and y components. The x-component of

F_1

is given by

F_{1x} = F_1 \cos(34^\circ)

and the y-component is

F_{1y} = F_1 \sin(34^\circ)

. Similarly, for

F_2

, we have

F_{2x} = F_2 \cos(-20^\circ)

and

F_{2y} = F_2 \sin(-20^\circ)

step 2

The net force in the x-direction can be expressed as:

F_{net,x} = F_{1x} + F_{2x} = 120 \cos(34^\circ) + F_2 \cos(-20^\circ)

The net force in the y-direction is:

F_{net,y} = F_{1y} + F_{2y} - mg = 120 \sin(34^\circ) + F_2 \sin(-20^\circ) - 40 \cdot 9.81

step 3

Next, we calculate the acceleration of the box using the kinematic equation. The box moves 50.0 meters in 8.00 seconds, starting from rest. The acceleration

a

can be calculated using:

a = \frac{2d}{t^2} = \frac{2 \cdot 50.0}{(8.00)^2} = \frac{100.0}{64} = 1.5625 \, \text{m/s}^2

step 4

Now, using Newton's second law, we can relate the net force to the mass and acceleration:

F_{net,x} = m \cdot a = 40.0 \cdot 1.5625 = 62.5 \, \text{N}

Setting the x-component equation equal to this value allows us to solve for

F_2

120 \cos(34^\circ) + F_2 \cos(-20^\circ) = 62.5

step 5

After calculating

F_2

, we can find the coefficient of kinetic friction

\mu_k

using the equation:

F_{friction} = \mu_k \cdot N

where

N

is the normal force. The normal force can be found from the y-component equation. Rearranging gives us:

\mu_k = \frac{F_{friction}}{N}

Answer

The magnitude of force

F_2

is approximately 36.5 N, and the coefficient of kinetic friction is approximately 0.25.

Key Concept

Understanding the components of forces and applying Newton's laws is crucial in solving problems involving motion and friction.

Explanation

The solution involves breaking down the forces into components, calculating acceleration, and using Newton's second law to find unknown forces and coefficients of friction.