# An object with a mass of #12# #kg# is on a surface with a kinetic friction coefficient of # 1 #. How much force is necessary to accelerate the object horizontally at #7 # #ms^-2#?

The total force is composed of two pieces: the force required to accelerate the mass and the force required to overcome friction.

The total force is made up of two different forces.

The second part is the frictional force:

To find the total force, just add these two forces:

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To calculate the force necessary to accelerate the object horizontally at 7 m/s^2, you need to consider both the force required to overcome kinetic friction and the force required to accelerate the object.

The force required to overcome kinetic friction is given by the equation: ( F_{friction} = \mu_k \times N ), where ( \mu_k ) is the coefficient of kinetic friction and ( N ) is the normal force.

The normal force is equal to the weight of the object, which is ( mg ), where ( m ) is the mass of the object and ( g ) is the acceleration due to gravity (( 9.8 , m/s^2 )).

So, ( N = mg = 12 , kg \times 9.8 , m/s^2 ).

Substituting the values into the equation for friction, we get:

( F_{friction} = 1 \times (12 , kg \times 9.8 , m/s^2) ).

Now, the force required to accelerate the object is given by the equation: ( F_{acceleration} = ma ), where ( m ) is the mass of the object and ( a ) is the acceleration.

Substituting the values into the equation for acceleration, we get:

( F_{acceleration} = 12 , kg \times 7 , m/s^2 ).

Now, add the force required to overcome friction and the force required for acceleration to get the total force:

( F_{total} = F_{friction} + F_{acceleration} ).

Calculate the values and add them together to find the total force.

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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 #4 kg# is lying still on a surface and is compressing a horizontal spring by #1/4 m#. If the spring's constant is #3 (kg)/s^2#, what is the minimum value of the surface's coefficient of static friction?
- An object, previously at rest, slides #3 m# down a ramp, with an incline of #pi/8 #, and then slides horizontally on the floor for another #4 m#. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?
- An object with a mass of # 5 kg# is on a ramp at an incline of #pi/8 #. If the object is being pushed up the ramp with a force of # 4 N#, what is the minimum coefficient of static friction needed for the object to remain put?
- An object, previously at rest, slides #8 m# down a ramp, with an incline of #pi/4 #, and then slides horizontally on the floor for another #6 m#. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?
- If a #2 kg# object moving at #10 m/s# slows down to a halt after moving #1 m#, what is the friction coefficient of the surface that the object was moving over?

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