A person pulls with a force of 40 N on two boxes at rest. The top box is 3.0 kg and the bottom box is 6.0 kg. The coefficient of kinetic friction between the floor and the larger box is .20 and the angle is 30 degrees.
a) if the smaller box doesn't slide on top of the larger box, what is the acceleration of the larger box?
b) Given your answer to part a, what must be the minimum value of the coefficient of static friction between the smaller box and the larger box?
To find the acceleration of the larger box and the minimum value of the coefficient of static friction between the smaller box and the larger box, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
Let's break down the problem step by step:
a) To find the acceleration of the larger box:
- First, determine the net force acting on the system. Since there are no external forces acting vertically on the two boxes, only the horizontal forces are considered.
- The top box is not sliding on top of the larger box, which means there is static friction between them. The static friction opposes the applied force, so the net horizontal force becomes the difference between the applied force and the force of friction.
- The force of friction can be calculated using the equation: force of friction = coefficient of friction * normal force.
- The normal force on the larger box is equal to its weight, which is given by: normal force = mass of the larger box * acceleration due to gravity.
- The net horizontal force can now be calculated as: net force = applied force - force of friction.
Once we have the net force, we can calculate the acceleration using Newton's second law:
- net force = mass of the system * acceleration.
- Rearranging the equation, we find: acceleration = net force / mass of the system.
b) To find the minimum value of the coefficient of static friction between the smaller box and the larger box:
- Assuming the smaller box is not sliding on top of the larger box, the force of static friction between them will prevent any relative motion.
- The force of static friction can be calculated using the equation: force of static friction = coefficient of static friction * normal force.
- We can set up an inequality to find the minimum value of the coefficient of static friction, where the force of static friction is greater than or equal to the force required to start the smaller box moving.
- The force required to start the smaller box moving can be calculated as: force required = mass of the smaller box * acceleration.
Let's plug in the given values and calculate the answers:
a) Calculate the acceleration of the larger box:
- Mass of the larger box = 3.0 kg
- Applied force = 40 N
- Coefficient of kinetic friction = 0.20
- Angle = 30 degrees
First, calculate the normal force on the larger box:
normal force = mass of the larger box * acceleration due to gravity
normal force = 3.0 kg * 9.8 m/s^2 = 29.4 N
Next, calculate the force of friction:
force of friction = coefficient of kinetic friction * normal force
force of friction = 0.20 * 29.4 N = 5.88 N
Calculate the net horizontal force:
net force = applied force - force of friction
net force = 40 N - 5.88 N = 34.12 N
Finally, calculate the acceleration of the larger box:
acceleration = net force / mass of the system
acceleration = 34.12 N / (3.0 kg + 6.0 kg) = 3.79 m/s^2
Therefore, the acceleration of the larger box is 3.79 m/s^2.
b) Calculate the minimum value of the coefficient of static friction between the smaller box and the larger box:
Assuming the smaller box is not sliding on top of the larger box, the force of static friction must be equal to or greater than the force required to start the smaller box moving.
force required = mass of the smaller box * acceleration
force required = 3.0 kg * 3.79 m/s^2 = 11.37 N
The force of static friction is given by: force of static friction = coefficient of static friction * normal force.
Since the force of static friction must be greater than or equal to the force required to start the smaller box, we have:
coefficient of static friction * normal force ≥ force required
Substituting the values:
coefficient of static friction * 29.4 N ≥ 11.37 N
Solving for the coefficient of static friction:
coefficient of static friction ≥ 11.37 N / 29.4 N
coefficient of static friction ≥ 0.39
Therefore, the minimum value of the coefficient of static friction between the smaller box and the larger box is 0.39.