# An object with a mass of #7 kg# is pushed along a linear path with a kinetic friction coefficient of #u_k(x)= 1+x^2-xcos(x) #. How much work would it take to move the object over #x in [pi, 4pi], where x is in meters?

The work done is

Integration between components

The completed work is

The force of friction is

The completed work is

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To calculate the work done to move the object over the given interval, we need to integrate the product of the force of friction and the distance traveled with respect to x:

[ W = \int_{\pi}^{4\pi} F_{\text{friction}}(x) , dx ]

Given that ( F_{\text{friction}}(x) = \mu_k(x) \cdot N ), where ( N ) is the normal force, and ( \mu_k(x) = 1 + x^2 - x\cos(x) ), we can find the normal force as ( N = mg ), where ( m ) is the mass of the object (7 kg) and ( g ) is the acceleration due to gravity (9.8 m/s²).

Now, we can integrate:

[ W = \int_{\pi}^{4\pi} (1 + x^2 - x\cos(x)) \cdot mg , dx ]

[ W = \int_{\pi}^{4\pi} (1 + x^2 - x\cos(x)) \cdot (7 \times 9.8) , dx ]

[ W = 68 \int_{\pi}^{4\pi} (1 + x^2 - x\cos(x)) , dx ]

After performing the integration over the given interval, we will obtain the work done to move the object over that distance.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- An object with a mass of # 5 kg# is lying on a surface and is compressing a horizontal spring by #20 cm#. If the spring's constant is # 4 (kg)/s^2#, what is the minimum value of the surface's coefficient of static friction?
- A truck pulls boxes up an incline plane. The truck can exert a maximum force of #5,200 N#. If the plane's incline is #(5 pi )/8 # and the coefficient of friction is #4/3 #, what is the maximum mass that can be pulled up at one time?
- If an object is moving at #12 m/s# over a surface with a kinetic friction coefficient of #u_k=2 /g#, how far will the object continue to move?
- An object with a mass of #2 kg# is acted on by two forces. The first is #F_1= <4 N , 6 N># and the second is #F_2 = < 1 N, -4 N>#. What is the object's rate and direction of acceleration?
- How fast will an object with a mass of #6 kg# accelerate if a force of #24 N# is constantly applied to it?

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