A child leaves her book bag on a slide. The bag, which is at the top of the slide, starts from rest and reaches the bottom in 1.63 s. The mass of the book bag is 2.55 kg, the length of the slide is 2.90 m and the angle of incline is 35.0°. (Assume the +x-axis to be parallel to and down the slide.)
(a) What is the friction force acting on the bag?
(b) What is the coefficient of kinetic friction between the bag and the slide?
To find the friction force acting on the bag, we need to first calculate the net force acting on it.
Step 1: Find the gravitational force acting on the bag.
The gravitational force is given by the equation:
F_gravity = m * g
where m is the mass of the bag and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting the given values:
F_gravity = 2.55 kg * 9.8 m/s^2
F_gravity = 24.99 N
Step 2: Find the parallel component of the gravitational force.
The parallel component of the gravitational force is given by the equation:
F_parallel = F_gravity * sin(angle of incline)
Substituting the given values:
F_parallel = 24.99 N * sin(35.0°)
F_parallel = 14.31 N
Step 3: Find the net force acting on the bag.
The net force is given by the equation:
F_net = m * a
where a is the acceleration of the bag.
Since the bag starts from rest and reaches the bottom of the slide in 1.63 s, we can calculate the acceleration using the equation:
a = 2 * (distance / time^2)
Substituting the given values:
a = 2 * (2.90 m / (1.63 s)^2)
a = 2.23 m/s^2
Substituting into the net force equation:
F_net = 2.55 kg * 2.23 m/s^2
F_net = 5.69 N
Step 4: Find the friction force.
The friction force can be calculated using the equation:
F_friction = F_net - F_parallel
Substituting the given values:
F_friction = 5.69 N - 14.31 N
F_friction = -8.62 N
(a) The friction force acting on the bag is -8.62 N, which implies that the direction of the friction force is opposite to the motion of the bag.
To find the coefficient of kinetic friction, we can use the equation:
μ_kinetic = F_friction / F_normal
where F_normal is the normal force.
Step 5: Find the normal force.
The normal force is given by the equation:
F_normal = F_gravity * cos(angle of incline)
Substituting the given values:
F_normal = 24.99 N * cos(35.0°)
F_normal = 20.44 N
Step 6: Find the coefficient of kinetic friction.
μ_kinetic = -8.62 N / 20.44 N
(b) The coefficient of kinetic friction between the bag and the slide is approximately -0.42.
To find the answers to these questions, we need to analyze the forces acting on the book bag as it slides down the incline. Let's break it down step by step:
Step 1: Calculate the gravitational force:
The gravitational force acting on the book bag can be calculated using the equation F_gravity = m * g, where m is the mass of the book bag and g is the acceleration due to gravity (approximately 9.8 m/s²). In this case, m = 2.55 kg, so the gravitational force is F_gravity = 2.55 kg * 9.8 m/s².
Step 2: Calculate the component of the gravitational force down the incline:
Since the angle of incline is given, we need to find the component of the gravitational force acting down the incline. This can be done using the equation F_gravity_parallel = F_gravity * sin(θ), where θ is the angle of incline (35.0°).
Step 3: Calculate the net force along the incline:
The net force acting on the book bag along the incline is the difference between the component of the gravitational force down the incline and the frictional force. The equation for the net force is F_net = F_gravity_parallel - F_friction.
Step 4: Calculate the acceleration along the incline:
Using Newton's second law, F = m * a, we can rearrange the equation to find the acceleration a = F_net / m.
Step 5: Calculate the frictional force:
The frictional force can be determined using the equation F_friction = μ * F_normal, where μ is the coefficient of kinetic friction and F_normal is the normal force. In this case, the normal force is equal to the component of the gravitational force perpendicular to the incline surface, which can be calculated using the equation F_normal = F_gravity * cos(θ).
Step 6: Calculate the coefficient of kinetic friction:
Finally, we can find the coefficient of kinetic friction by rearranging the frictional force equation to μ = F_friction / F_normal.
Now, let's plug in the given values and solve the equations:
Step 1: Calculate the gravitational force:
F_gravity = 2.55 kg * 9.8 m/s²
Step 2: Calculate the component of the gravitational force down the incline:
F_gravity_parallel = F_gravity * sin(θ)
Step 3: Calculate the net force along the incline:
F_net = F_gravity_parallel - F_friction
Step 4: Calculate the acceleration along the incline:
a = F_net / m
Step 5: Calculate the frictional force:
F_friction = μ * F_normal, where F_normal = F_gravity * cos(θ)
Step 6: Calculate the coefficient of kinetic friction:
μ = F_friction / F_normal
Plug in the given values and solve these equations to find the answers.