A 570 kg truck is carrying a large 1.9 kg box on its cargo deck when it starts to accelerate. If the coefficient of static friction between the box and the truck’s deck is 0.26, what is the maximum acceleration, in m/s 2, that the truck can have without letting the box slip?
Well, well, it seems we have a truck trying to juggle a box! Let's calculate the maximum acceleration without letting that box slip, shall we?
To find the maximum acceleration, we need to consider the friction force between the box and the truck's deck. The maximum static friction force can be calculated using the formula:
F_static_max = μ * m * g
where:
μ is the coefficient of static friction (0.26),
m is the mass of the box (1.9 kg), and
g is the acceleration due to gravity (approximately 9.8 m/s²).
Plugging in the values, we get:
F_static_max = 0.26 * 1.9 kg * 9.8 m/s²
Now, we know that the maximum static friction force between the box and the truck is equal to the mass of the box multiplied by its acceleration. So, we can set up an equation:
m * a_max = F_static_max
Let's solve for a_max:
1.9 kg * a_max = 0.26 * 1.9 kg * 9.8 m/s²
Simplifying:
a_max = (0.26 * 1.9 kg * 9.8 m/s²) / 1.9 kg
Calculating that, we find:
a_max ≈ 0.2667 m/s²
So, the maximum acceleration the truck can have without letting the box slip is approximately 0.2667 m/s². Keep that box juggling under control, or it might just fall off the truck!
To determine the maximum acceleration without letting the box slip, we need to calculate the maximum frictional force that can be exerted on the box.
The frictional force is given by the equation:
frictional force = coefficient of static friction × normal force
The normal force is equal to the weight of the box, which can be calculated as:
normal force = mass of the box × acceleration due to gravity
Given:
Mass of the box (m) = 1.9 kg
Coefficient of static friction (μ) = 0.26
Acceleration due to gravity (g) = 9.8 m/s^2
Therefore, the normal force is:
normal force = 1.9 kg × 9.8 m/s^2 = 18.62 N
Now, we can calculate the maximum frictional force:
frictional force = 0.26 × 18.62 N = 4.8332 N
Since the frictional force is equal to the maximum force of static friction, we can use Newton's second law to determine the maximum acceleration:
frictional force = mass of the truck × acceleration
Given:
Mass of the truck = 570 kg
570 kg × acceleration = 4.8332 N
Solving for acceleration:
acceleration = 4.8332 N / 570 kg
acceleration ≈ 0.0085 m/s^2
Therefore, the maximum acceleration that the truck can have without letting the box slip is approximately 0.0085 m/s^2.
To determine the maximum acceleration without letting the box slip, we need to compare the force of static friction between the box and the truck's deck to the maximum possible frictional force.
The maximum frictional force between two surfaces can be calculated using the equation:
F_friction = μ * F_normal
Where:
F_friction is the frictional force
μ is the coefficient of static friction
F_normal is the normal force (equal to the weight of the box)
First, let's calculate the weight of the box:
Weight = mass * gravity
Where:
mass = 1.9 kg (given)
gravity = 9.8 m/s^2 (approximate acceleration due to gravity)
Weight = 1.9 kg * 9.8 m/s^2 = 18.62 N
Now, we can calculate the maximum frictional force:
F_friction_max = μ * F_normal
F_friction_max = 0.26 * 18.62 N = 4.8372 N
Since the maximum force of static friction is equal to the maximum frictional force, we can use the equation:
F_friction_max = mass * acceleration_max
Solving for acceleration_max:
acceleration_max = F_friction_max / mass
acceleration_max = 4.8372 N / 570 kg = 0.0085 m/s^2
Therefore, the maximum acceleration the truck can have without letting the box slip is approximately 0.0085 m/s^2.
Fn = mg
Fs = coeff of static fiction(Fn)
a=Fs/m
note. use mass of box only