A book of mass 7 kg rests on a plank. You tilt one end of the plank and slowly increase the angle of the tilt. The coefficient of static friction between the book and the plank is 0.40. What is the maximum angle of tilt for which the book will remain stationary and not slide down the plank?
m*g = 7 * 9.8 = 68.6 N. = Wt. of book.
Fp = 68.6*sin A = Force parallel to the
plank.
Fn = 68.6*Cos A = Force perpendicular to
the plank.
Fs = u*Fn = 0.4*68.6*Cos A
Fp-Fs = m*a
68.6*sin A - 0.4*68.6*Cos A = m*0 = 0
68.6*Sin A = 0.4*68.6*Cos A
Sin A = 0.4*Cos A
Divide both sides by Cos A:
Sin A/Cos A = 0.4
Replace Sin A/Cos A with Tan A:
Tan A = 0.4
A = 21.8o, Max.
Why did the book go to college? Because it wanted to get degrees of static friction! But let's get serious for a moment. To find the maximum angle of tilt, we need to consider the force of gravity pulling the book down and the frictional force keeping it in place.
The force of gravity can be calculated by multiplying the mass of the book (7 kg) by the acceleration due to gravity (9.8 m/s^2), which gives us a value of 68.6 N.
The frictional force can be calculated by multiplying the normal force (which is equal to the force of gravity in this case) by the coefficient of static friction, which gives us a value of 27.4 N.
Now, we can use trigonometry to find the maximum angle of tilt. The maximum angle of tilt is the angle at which the force component of gravity perpendicular to the plank (mg * sin(angle)) is equal to the frictional force. So, we can set up the following equation:
mg * sin(angle) = frictional force
Plugging in the values, we get:
(7 kg * 9.8 m/s^2) * sin(angle) = 27.4 N
Simplifying the equation, we find:
sin(angle) = 0.4
To find the angle itself, we take the inverse sine of both sides:
angle = arcsin(0.4)
Using a calculator, we find that the maximum angle of tilt is approximately 23.6 degrees. So, the book will remain stationary and not slide down the plank as long as the angle of tilt is less than or equal to 23.6 degrees.
To find the maximum angle of tilt for which the book will remain stationary and not slide down the plank, we need to consider the forces acting on the book.
There are two forces acting on the book: the gravitational force (weight) pulling it downwards, and the static friction force between the book and the tilted plank opposing its movement.
Let's consider the forces in the vertical direction. The weight of the book is given by the equation:
Weight = mass * gravitational acceleration
Weight = 7 kg * 9.8 m/s^2 = 68.6 N
Next, we consider the forces in the horizontal direction. The static friction force can be calculated using the equation:
Static friction force = coefficient of static friction * normal force
The normal force is the component of the weight perpendicular to the inclined plane. It can be calculated using trigonometry:
Normal force = Weight * cos(angle of tilt)
Finally, the maximum angle of tilt at which the book will not slide is when the static friction force is equal to its maximum value. The maximum static friction force can be calculated using the equation:
Maximum static friction force = coefficient of static friction * normal force
To find the maximum angle of tilt, we need to find the angle at which the maximum static friction force is equal to the weight of the book. So we have:
Maximum static friction force = weight
coefficient of static friction * normal force = 68.6 N
coefficient of static friction * (Weight * cos(angle of tilt)) = 68.6 N
0.4 * (7 kg * 9.8 m/s^2 * cos(angle of tilt)) = 68.6 N
0.4 * 68.6 N = 27.44 N * cos(angle of tilt)
0.4 = cos(angle of tilt)
Angle of tilt = arccos(0.4)
Angle of tilt = 66.4 degrees (rounded to 1 decimal place)
Therefore, the maximum angle of tilt for which the book will remain stationary and not slide down the plank is approximately 66.4 degrees.
To find the maximum angle of tilt at which the book will remain stationary and not slide down the plank, we need to consider the forces acting on the book.
Let's break down the forces:
1. The gravitational force (Fg) acting vertically downwards with a magnitude of m * g, where m is the mass of the book and g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. The normal force (Fn) acting perpendicular to the plank. It counteracts the gravitational force and has the same magnitude as Fg.
3. The static friction force (Fs) acting parallel to the plank, opposing the motion of the book. It depends on the coefficient of static friction (μs) and the normal force (Fn).
The maximum angle of tilt occurs when the component of the gravitational force parallel to the plank equals the maximum static friction force. This condition can be expressed as:
Fs(max) = μs * Fn,
where Fs(max) is the maximum static friction force.
Since Fn equals the magnitude of Fg, we have:
Fs(max) = μs * Fg.
Substituting the values:
Fs(max) = 0.40 * (7 kg) * (9.8 m/s^2)
Fs(max) = 27.44 N.
Now, we can determine the angle of tilt at which the gravitational force parallel to the plank equals the maximum static friction force:
Fg_parallel = Fg * sin(θ),
where θ is the angle of tilt.
Setting Fg_parallel equal to Fs(max):
Fg * sin(θ) = Fs(max),
(7 kg) * (9.8 m/s^2) * sin(θ) = 27.44 N.
Now we can solve for θ:
sin(θ) = 27.44 N / (7 kg * 9.8 m/s^2),
sin(θ) ≈ 0.3904.
Taking the inverse sine (sin^(-1)) of both sides to get θ:
θ ≈ sin^(-1)(0.3904),
θ ≈ 23.5 degrees.
Therefore, the maximum angle of tilt for which the book will remain stationary and not slide down the plank is approximately 23.5 degrees.