A 35 kg box rest in the bed of a truck. The coefficient of static friction between the box and the truck bed is 0.30 and the coefficient of friction of kinetic friction is 0.10.
A) Calculate the maximum acceleration the truck can have before the box slides backwards.
B) The truck just exceeds the maximum acceleration that prevents the box from sliding backwards. Calculate the acceleration of the box.
Wb = m*g = 35kg * 9.8N/kg = 343 N. = Wt.
of box.
Fb = 343N @ 0o.
Fp=343*sin(0) = 0.=Force parallel to bed
Fv = 343*cos(0) = 343 N. = Force perpendicular to bed.
Fs = u*Fv = 0.3*343 = 102.9 N. = Force of static friction.
A. Fn = Fp - Fs = m*a.
0 - 102.9 = 35*a
qa = -2.94 m/s^2.
B. Fk = u*Fv = 0.1*343 = 34.3 N.
Fp - Fk = m*a.
0-34.3 = 35*a
a = -0.98 m/s^2.
A) Maximum acceleration can be determined by using the equation F_friction = μ_s * F_normal, where F_friction is the force of friction, μ_s is the coefficient of static friction, and F_normal is the normal force.
The normal force on the box is equal to its weight, which is given by F_normal = m * g, where m is the mass of the box and g is the acceleration due to gravity.
For the maximum acceleration, the force of friction should be equal to the maximum static friction force, so F_friction = μ_s * F_normal.
Substituting the equations, we have μ_s * m * g = μ_s * m * g * a_max, where a_max is the maximum acceleration.
Simplifying, we get a_max = g * μ_s.
Given that the coefficient of static friction is 0.30 and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the maximum acceleration:
a_max = 9.8 m/s^2 * 0.30 = 2.94 m/s^2.
Therefore, the maximum acceleration the truck can have before the box slides backwards is 2.94 m/s^2.
B) Once the box starts sliding, the friction force changes from static to kinetic friction. The force of kinetic friction is given by F_friction = μ_k * F_normal, where μ_k is the coefficient of kinetic friction.
To calculate the acceleration of the box, we can use Newton's second law, F_net = m * a, where F_net is the net force acting on the box.
Since the box is sliding, the net force can be written as F_net = F_applied - F_friction, where F_applied is the applied force.
Substituting the equation for F_friction, we have F_net = F_applied - μ_k * F_normal.
Since the box is sliding, the applied force equals the force of kinetic friction, so F_applied = μ_k * F_normal.
Substituting again, we get F_net = μ_k * F_normal - μ_k * F_normal = 0.
Therefore, the net force acting on the box is zero, and according to Newton's second law, this means the acceleration of the box is also zero.
So, the acceleration of the box is 0 m/s^2.
A) To calculate the maximum acceleration the truck can have before the box slides backward, we need to determine the maximum static friction force that can be exerted between the box and the truck bed.
The formula for static friction is:
Fs = μs * N
where Fs is the static friction force, μs is the coefficient of static friction, and N is the normal force.
The normal force is the force exerted by the truck bed on the box, which is equal to the weight of the box. The weight of the box can be calculated using the formula:
W = m * g
where W is the weight, m is the mass of the box, and g is the acceleration due to gravity.
Given that the mass of the box is 35 kg and the coefficient of static friction is 0.30, we can calculate the maximum static friction force as follows:
Fs = 0.30 * m * g
Now, we need to convert the mass of the box to weight:
W = 35 kg * 9.8 m/s^2 = 343 N
Substituting the values into the formula, we get:
Fs = 0.30 * 343 N = 102.9 N
The maximum static friction force is 102.9 N.
To find the maximum acceleration, we can use Newton's second law of motion:
Fnet = m * a
where Fnet is the net force, m is the mass of the box, and a is the acceleration.
Since there are only two forces acting on the box: the static friction force and the gravitational force, the net force is given by:
Fnet = Fs - W
Substituting the values, we have:
Fnet = 102.9 N - 343 N = -240.1 N
Since the box is on the verge of sliding backward, the net force is equal to the maximum static friction force:
Fnet = Fs
Therefore, we can rewrite the equation as:
Fs = m * a
Substituting the values, we get:
102.9 N = 35 kg * a
Solving for a, we have:
a = 102.9 N / 35 kg = 2.94 m/s^2
Therefore, the maximum acceleration the truck can have before the box slides backward is 2.94 m/s^2.
B) The acceleration of the box when the truck just exceeds the maximum acceleration that prevents the box from sliding backward can be calculated using the formula:
Fnet = m * a
where Fnet is the net force, m is the mass of the box, and a is the acceleration.
In this case, the net force is given by:
Fnet = μk * N
where μk is the coefficient of kinetic friction and N is the normal force.
The normal force is still equal to the weight of the box, which is 343 N.
Substituting the values, we have:
Fnet = 0.10 * 343 N = 34.3 N
Since the net force is equal to the weight of the box, we can write the equation as:
34.3 N = 35 kg * a
Solving for a, we have:
a = 34.3 N / 35 kg = 0.98 m/s^2
Therefore, the acceleration of the box is 0.98 m/s^2.
To solve these problems, we need to understand the concept of friction and how it relates to the forces acting on the box.
First, let's analyze the forces acting on the box when it is at rest in the bed of the truck:
1. The gravitational force (weight) acting vertically downwards, given by Fg = mg, where m is the mass of the box (35 kg) and g is the acceleration due to gravity (9.8 m/s^2).
2. The normal force (N) exerted by the truck bed on the box. This force is equal in magnitude but opposite in direction to the weight of the box (N = mg).
3. The static friction force (fs) acts horizontally between the box and the truck bed and prevents it from sliding. The maximum frictional force that can be exerted in the static case is given by the equation fs(max) = μsN, where μs is the coefficient of static friction.
Now, let's calculate the maximum acceleration the truck can have before the box slides backwards (question A):
Since the box is at rest, the sum of forces in the horizontal direction is zero. Therefore, the frictional force cancels out the force that would attempt to move the box. Mathematically, this can be expressed as:
fs = μsN = μsmg
To find the maximum acceleration (a_max) that the truck can have before the box slides, we need to determine when the frictional force is at its maximum. The maximum static frictional force is given by fs(max) = μsN.
Since fs = μsmg and fs = fs(max), we have μsmg = μsN.
Substituting the expression for N (N = mg), we get:
μsmg = μs(mg)
Simplifying, we find:
a_max = g
Therefore, the maximum acceleration the truck can have before the box slides backwards is equal to the acceleration due to gravity, which is approximately 9.8 m/s^2.
Now let's calculate the acceleration of the box when the truck just exceeds the maximum acceleration that prevents the box from sliding backwards (question B):
In this scenario, the frictional force is no longer strong enough to prevent the box from sliding. The box will experience kinetic friction instead of static friction.
The kinetic friction force (fk) is given by the equation fk = μkN, where μk is the coefficient of kinetic friction.
The acceleration of the box (a_box) can be found by applying Newton's second law, which states that the net force is equal to the mass of the object multiplied by its acceleration (F_net = ma).
The net force acting on the box is given by:
F_net = fk - Fg
Substituting the values, we get:
ma_box = μkN - mg
Substituting N = mg and rearranging for acceleration, we have:
a_box = μk(g)
Therefore, the acceleration of the box when the truck just exceeds the maximum acceleration is equal to the acceleration due to gravity multiplied by the coefficient of kinetic friction (a_box = μk(g)).