# A block weighing #15 kg# is on a plane with an incline of #pi/3# and friction coefficient of #1/10#. How much force, if any, is necessary to keep the block from sliding down?

I'll assume that *static* friction.

We're asked to find if any force is required (and if there is, what is it) to keep the block stationary and prevent it from sliding down.

We have our relationship for the friction force

Here,

The normal force

The quantity

This represents the *maximum* static friction force that prevents the object from sliding.

Taking the positive *down* the incline, we have

Since this number is greater than the allowed static friction force of

This force is

directed up the incline.

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The force is

Resolving in the direction parallel to the plane

Let the force necessary to keep the block from sliding down be

The force of friction is

The component of the weight is

Therefore,

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To keep the block from sliding down the incline, a force equal to or greater than the force of friction needs to be applied. The force of friction can be calculated using the formula:

( F_{friction} = \mu \cdot N )

where ( \mu ) is the coefficient of friction and ( N ) is the normal force. The normal force can be calculated as:

( N = mg \cdot \cos(\theta) )

where ( m ) is the mass of the block, ( g ) is the acceleration due to gravity, and ( \theta ) is the angle of the incline.

Substituting the given values:

( N = (15 kg) \cdot (9.8 m/s^2) \cdot \cos(\pi/3) )

( N = 15 kg \cdot 9.8 m/s^2 \cdot 0.5 )

( N = 73.5 N )

Now, substituting the value of ( N ) into the formula for friction:

( F_{friction} = (1/10) \cdot 73.5 N )

( F_{friction} = 7.35 N )

Therefore, a force of at least ( 7.35 , \text{N} ) is necessary to keep the block from sliding down the incline.

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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.

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