# An object with a mass of #5 kg# is pushed along a linear path with a kinetic friction coefficient of #u_k(x)= e^x-2x+3 #. How much work would it take to move the object over #x in [3, 4], where x is in meters?

where

The necessary force would need to be equal to (or greater than, but we're looking for the minimum value) the retarding friction force, so

And we also plug in the above coefficient of kinetic friction equation:

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[ W = \int_{x_1}^{x_2} F_{\text{net}} , dx ] [ F_{\text{net}} = m \cdot a ] [ a = \frac{dv}{dt} = \frac{dx}{dt} \cdot \frac{dv}{dx} = v \cdot \frac{dv}{dx} ] [ F_{\text{net}} = m \cdot v \cdot \frac{dv}{dx} - \mu_k \cdot m \cdot g ] [ W = \int_{x_1}^{x_2} \left( m \cdot v \cdot \frac{dv}{dx} - \mu_k \cdot m \cdot g \right) , dx ] [ W = \int_{3}^{4} \left( 5 \cdot v \cdot \frac{dv}{dx} - (e^x - 2x + 3) \cdot 5 \cdot 9.8 \right) , dx ] [ W \approx 377.6 , \text{J} ]

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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 # 15 kg# is lying on a surface and is compressing a horizontal spring by #10 cm#. If the spring's constant is # 8 (kg)/s^2#, what is the minimum value of the surface's coefficient of static friction?
- If the length of a #17 cm# spring increases to #63 cm# when a #3 kg# weight is hanging from it, what is the spring's constant?
- If an object is moving at #3 m/s# over a surface with a kinetic friction coefficient of #u_k=5 /g#, how far will the object continue to move?
- If an object is moving at #150 m/s# over a surface with a kinetic friction coefficient of #u_k=15 /g#, how far will the object continue to move?
- An object, previously at rest, slides #4 m# down a ramp, with an incline of #pi/12 #, and then slides horizontally on the floor for another #3 m#. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?

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