A book whose mass is 2kg is projected up a long 37degree inclined plane with initial velocity, 24m/s. The coefficient of kinetic friction between the plane and the book is 0.25. Use g= 10m/s^2.
a) Find friction force.
b) How much time does it take before the book comes to a rest?
To find the friction force acting on the book, we need to consider the force of gravity and the normal force acting on the book.
a) Friction Force:
The force of gravity acting on the book can be calculated using the formula:
F_gravity = m * g
where m = mass of the book = 2 kg
and g = acceleration due to gravity = 10 m/s^2
F_gravity = 2 kg * 10 m/s^2
= 20 N
The normal force acting on the book can be calculated using the formula:
F_normal = m * g * cos(theta)
where theta = angle of inclination = 37 degrees
F_normal = 2 kg * 10 m/s^2 * cos(37 degrees)
= (2 kg * 10 m/s^2 * 0.7986)
= 15.972 N
The friction force can be calculated using the formula:
F_friction = coefficient of kinetic friction * F_normal
where coefficient of kinetic friction = 0.25
F_friction = 0.25 * 15.972 N
= 3.993 N
Therefore, the friction force acting on the book is approximately 3.993 N.
b) Time taken to come to rest:
To find the time it takes for the book to come to a complete rest, we can use the concept of deceleration.
The net force acting on the book in the direction opposite to its motion is given by:
F_net = F_gravity - F_friction
F_net = 20 N - 3.993 N
= 16.007 N
The deceleration (a) can be calculated using Newton's second law of motion:
F_net = m * a
where m = mass of the book = 2 kg
16.007 N = 2 kg * a
a = 8.0035 m/s^2
Using the equation of motion:
v = u + a * t
where v = final velocity = 0 m/s (since the book comes to rest)
u = initial velocity = 24 m/s
a = acceleration = -8.0035 m/s^2 (negative sign indicates deceleration)
t = time taken to come to rest (unknown)
0 = 24 m/s + (-8.0035 m/s^2) * t
-24 m/s = -8.0035 m/s^2 * t
t = 24 m/s / 8.0035 m/s^2
t ≈ 3 seconds
Therefore, it takes approximately 3 seconds for the book to come to a rest.