The 3.42 kg physics book shown is connected by a string to a 518.0 g coffee cup. The book is given a push up the slope and released with a speed of 2.73 m/s. The coefficients of friction are μs = 0.490 and μk = 0.185. What is the acceleration of the book if the slope is inclined at 27.4°?
ibid.
To find the acceleration of the book, we need to calculate the net force acting on it.
First, let's find the normal force, which is the force perpendicular to the slope. The normal force (Fn) can be calculated using the following formula:
Fn = m * g * cos(theta)
where m is the mass of the book, g is the acceleration due to gravity, and theta is the angle of the slope.
Given:
Mass of the book (m) = 3.42 kg
Acceleration due to gravity (g) = 9.8 m/s^2
Angle of the slope (theta) = 27.4°
Fn = 3.42 kg * 9.8 m/s^2 * cos(27.4°)
Fn = 3.42 kg * 9.8 m/s^2 * 0.8828
Fn = 29.205 N (rounded to three decimal places)
Next, let's calculate the force of friction acting on the book.
The force of friction can be calculated using the following formula:
Ff = μ * Fn
where μ is the coefficient of friction and Fn is the normal force.
Given:
Coefficient of static friction (μs) = 0.490
Coefficient of kinetic friction (μk) = 0.185
Normal force (Fn) = 29.205 N
For static friction:
Fs = μs * Fn
Fs = 0.490 * 29.205 N
Fs = 14.310 N (rounded to three decimal places)
For kinetic friction:
Fk = μk * Fn
Fk = 0.185 * 29.205 N
Fk = 5.403 N (rounded to three decimal places)
Now, let's calculate the net force acting on the book.
The net force (Fnet) can be calculated using the following formula:
Fnet = m * a
where m is the mass of the book and a is the acceleration.
Given:
Mass of the book (m) = 3.42 kg
Fnet = 3.42 kg * a
Since the book is pushed up the slope, we need to consider the force of friction in the opposite direction:
Fnet = Fs - Fk = 14.310 N - 5.403 N
Fnet = 8.907 N (rounded to three decimal places)
Setting this equal to the net force:
8.907 N = 3.42 kg * a
Now, solve for the acceleration a:
a = 8.907 N / 3.42 kg
a ≈ 2.601 m/s²
Therefore, the acceleration of the book is approximately 2.601 m/s².
To find the acceleration of the book, we need to calculate the net force acting on it along the incline.
First, let's find the force of gravity acting on the book. The force of gravity can be found using the formula:
Fg = m * g
where m is the mass of the book and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Given that the mass of the book is 3.42 kg, the force of gravity is:
Fg = 3.42 kg * 9.8 m/s^2 = 33.516 N
Next, we need to determine the force of friction acting on the book. The force of friction can be calculated using the equation:
Ff = μ * Fn
where μ is the coefficient of friction and Fn is the normal force.
The normal force is the component of the force of gravity perpendicular to the incline, which can be calculated using:
Fn = mg * cos(θ)
where θ is the angle of the slope (27.4°) and mg is the gravitational force.
Fn = (3.42 kg + 0.518 kg) * 9.8 m/s^2 * cos(27.4°) = 37.674 N
Now, let's calculate the force of friction:
Ff = μs * Fn = 0.490 * 37.674 N = 18.437 N
Since the book is initially at rest and has a speed of 2.73 m/s after being pushed, it means that the force of friction is kinetic rather than static. In this case, we need to use the kinetic friction coefficient (μk) instead of the static friction coefficient (μs).
Now, let's calculate the force of friction:
Ff = μk * Fn = 0.185 * 37.674 N = 6.976 N
The net force along the incline is given by:
Fnet = Fg * sin(θ) - Ff = 33.516 N * sin(27.4°) - 6.976 N = 8.001 N
Finally, we can calculate the acceleration of the book using Newton's second law:
Fnet = m * a
a = Fnet / m = 8.001 N / 3.42 kg = 2.342 m/s^2
Therefore, the acceleration of the book is approximately 2.342 m/s^2.