A trash can with a mass of 5 kg has a coefficient of static friction of μs = 0.80 and a coefficient of kinetic friction of μk = 0.22. (The Active Figure applies to a similar situation, but the values of the mass, coefficient of static friction and coefficient of kinetic friction may be different. So the values it gives may not be the same values you obtain from your calculations.)
(A) What is the maximum horizontal force Farrowbold that can be exerted without moving the trash can?
(B) Suppose this force is just barely exceeded. Determine the acceleration of the trash can as it moves.
To solve these questions, we can use Newton's laws of motion and the equations of friction.
(A) To determine the maximum horizontal force that can be exerted without moving the trash can, we need to consider the static friction. The formula for static friction is given by:
F_static = μs * N
where F_static is the static friction force, μs is the coefficient of static friction, and N is the normal force.
In this case, the normal force is equal to the weight of the trash can, which is given by:
N = m * g
where m is the mass of the trash can and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Plugging in the values:
m = 5 kg
μs = 0.80
we can calculate the normal force:
N = m * g
N = 5 kg * 9.8 m/s^2
N ≈ 49 N
Now we can calculate the maximum force:
F_static = μs * N
F_static = 0.80 * 49 N
F_static ≈ 39.2 N
Therefore, the maximum horizontal force (without moving the trash can) is approximately 39.2 Newtons (N).
(B) If the force exerted is just barely exceeded, then the trash can will start moving, and we need to consider the kinetic friction. The formula for kinetic friction is given by:
F_kinetic = μk * N
where F_kinetic is the kinetic friction force, μk is the coefficient of kinetic friction, and N again represents the normal force.
In this case, since the force is just exceeded, the applied force will be equal to the kinetic friction force:
F_applied = F_kinetic
Plugging in the values we already calculated for N:
N ≈ 49 N
and the given coefficient of kinetic friction:
μk = 0.22
we can determine the kinetic friction force:
F_kinetic = μk * N
F_kinetic = 0.22 * 49 N
F_kinetic ≈ 10.78 N
Now we can use Newton's second law to determine the acceleration:
F_net = m * a
where F_net is the net force (F_applied - F_kinetic), m is the mass of the trash can, and a is the acceleration.
Plugging in the values:
Mass (m) = 5 kg
F_applied = F_kinetic = 10.78 N
we can solve for the acceleration:
F_net = m * a
F_applied - F_kinetic = m * a
10.78 N - 10.78 N = 5 kg * a
0 = 5 kg * a
Since the net force is zero (as the force applied is just barely exceeded), the acceleration of the trash can is zero.
Therefore, the acceleration of the trash can as it moves, when the force just barely exceeds the maximum static friction, is zero.
To solve these questions, we will use the concepts of static and kinetic friction. The maximum horizontal force that can be exerted without moving the trash can is given by the equation:
Fmax = μs * N
where Fmax is the maximum force, μs is the coefficient of static friction, and N is the normal force. The normal force is equal to the weight of the object, which can be calculated using the formula:
N = m * g
where m is the mass and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Let's calculate the answers to the given questions:
(A) What is the maximum horizontal force Farrowbold that can be exerted without moving the trash can?
To find Fmax, we need to calculate N first using the formula N = m * g:
N = 5 kg * 9.8 m/s^2 = 49 N
Now, we can calculate Fmax using the formula Fmax = μs * N:
Fmax = 0.80 * 49 N = 39.2 N
Therefore, the maximum horizontal force that can be exerted without moving the trash can is 39.2 Newtons.
(B) Suppose this force is just barely exceeded. Determine the acceleration of the trash can as it moves.
In this case, the force being applied is greater than the maximum force (Fmax). Since the force is greater, we are dealing with kinetic friction instead of static friction.
The force of kinetic friction (Fk) can be calculated using the equation:
Fk = μk * N
where μk is the coefficient of kinetic friction.
Using the same value of N (49 N), we can calculate Fk:
Fk = 0.22 * 49 N = 10.78 N
Since the applied force exceeds the force of kinetic friction, the resulting net force is the difference between the applied force and the force of kinetic friction:
Net force = Fapplied - Fk = Fapplied - 10.78 N
The net force is also equal to the product of mass (m) and acceleration (a):
Net force = m * a
Setting these two expressions equal, we can solve for the acceleration:
Fapplied - 10.78 N = m * a
Substituting the given mass of 5 kg, we have:
Fapplied - 10.78 N = 5 kg * a
Remember that the given force, which is just barely exceeded, is the maximum force that can be exerted without moving the trash can. Therefore, the acceleration (a) in this scenario is 0 since the trash can is not moving.
So, the acceleration of the trash can as it moves when the force is just barely exceeded is 0 m/s^2.
weight = 5*9.81
max static friction force =5*9.81*.8
= 39.24 N
5*9.81 * .22 = 10.8 moving friction
39.24 - 10.8 = 5 a
a = 5.69 m/s^2