A 45 kg crate is placed on an inclined ramp. When the angle the ramp makes with the horizontal is increased to 29 degrees, the crate begins to slide downward.
What is the coefficient of static friction between the crate and the ramp?
At what angle does the crate begin to slide if its mass is doubled?
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Coefficient of static friction = 0.4877
To find the coefficient of static friction between the crate and the ramp, we can use the fact that the crate begins to slide when the force of gravity component parallel to the ramp exceeds the force of static friction.
Let's start by finding the force of gravity parallel to the ramp. To do this, we need to find the component of the weight of the crate that acts along the ramp. This can be done by multiplying the weight (mass × gravity) by the sine of the angle between the ramp and the horizontal:
Force of gravity parallel to the ramp = Weight × sin(angle)
In this case, the weight of the crate is given by mass × gravity, so we have:
Force of gravity parallel to the ramp = (mass × gravity) × sin(angle)
Substituting the given values, we have:
Force of gravity parallel to the ramp = (45 kg × 9.8 m/s^2) × sin(29°)
Next, we'll find the force of static friction. Since the crate is at the verge of sliding, the force of static friction is at its maximum value. The force of friction can be calculated using the equation:
Force of friction = coefficient of static friction × Normal force
where the Normal force is the force perpendicular to the ramp acting on the crate. The Normal force can be found by multiplying the weight of the crate by the cosine of the angle between the ramp and the horizontal:
Normal force = Weight × cos(angle)
Substituting the given values, we have:
Normal force = (mass × gravity) × cos(angle)
Now we can calculate the force of static friction:
Force of static friction = coefficient of static friction × (mass × gravity) × cos(angle)
Since the crate is at the verge of sliding, the force of static friction is equal to the force of gravity parallel to the ramp. Therefore, we can set these two forces equal to each other:
coefficient of static friction × (mass × gravity) × cos(angle) = (mass × gravity) × sin(angle)
Canceling out the mass and gravity, and rearranging the equation, we get:
coefficient of static friction = sin(angle) / cos(angle)
Using the given angle of 29°, we can calculate the coefficient of static friction:
coefficient of static friction = sin(29°) / cos(29°)
Now we can calculate the coefficient of static friction.
To find the angle at which the crate begins to slide when its mass is doubled, we can use the same equation:
coefficient of static friction × (mass × gravity) × cos(angle) = (mass × gravity) × sin(angle)
But this time, we need to use the doubled mass of the crate and solve for the angle:
coefficient of static friction × (2 × mass × gravity) × cos(angle) = (2 × mass × gravity) × sin(angle)
Dividing both sides of the equation by (2 × mass × gravity), we have:
coefficient of static friction × cos(angle) = sin(angle)
Dividing both sides of the equation by cos(angle), we get:
coefficient of static friction = sin(angle) / cos(angle)
Now, we can solve for the angle using the doubled mass and the given coefficient of static friction.