A student moves a box of books down the hall
by pulling on a rope attached to the box. The
student pulls with a force of 173 N at an angle of 27.5� above the horizontal. The box has a
mass of 27 kg, and μk between the box and
the floor is 0.2.
The acceleration of gravity is 9.81 m/s2 .
Find the acceleration of the box.
Answer in units of m/s2
Divide the net horizontal forward force by the mass.
[173cos27.5 -(M*g -173sin27.5)*ìk]/M = a
M = 27 kg
ìk = 0.2
g = 9.81
Solve for a.
-0.232 m/s^2
The force applied by the student can be resolved into two components: one parallel to the floor and one perpendicular to the floor.
The component of force parallel to the floor is given by:
F_parallel = F * cos(angle)
F_parallel = 173 N * cos(27.5º)
F_parallel ≈ 156.16 N
The force of friction acting on the box can be calculated using the equation:
Frictional force = coefficient of friction * normal force
Normal force = mass * gravitational acceleration
Normal force = 27 kg * 9.81 m/s^2
Normal force ≈ 264.87 N
Frictional force = 0.2 * 264.87 N
Frictional force ≈ 52.97 N
Since the box is moving, the applied force F_parallel must be greater than the force of friction. Therefore:
F_parallel > Frictional force
We can use Newton's second law to calculate the acceleration of the box:
F_parallel - Frictional force = mass * acceleration
156.16 N - 52.97 N = 27 kg * acceleration
103.19 N = 27 kg * acceleration
Acceleration = 103.19 N / 27 kg
Acceleration ≈ 3.82 m/s²
Therefore, the acceleration of the box is approximately 3.82 m/s².
To find the acceleration of the box, we need to 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.
First, we need to determine the horizontal and vertical components of the force pulling the box. The horizontal component can be found using the formula: Fx = F * cos(theta), where F is the applied force and theta is the angle above the horizontal.
Fx = 173 N * cos(27.5 degrees) = 155.25 N
The frictional force opposing the motion of the box can be found using the formula: Ff = μk * N, where μk is the coefficient of kinetic friction and N is the normal force.
The normal force N is equal to the weight of the box, which can be found using the formula: N = m * g, where m is the mass of the box and g is the acceleration due to gravity.
N = 27 kg * 9.81 m/s^2 = 264.87 N
Ff = 0.2 * 264.87 N = 52.97 N
Since the box is moving horizontally, the net force in the horizontal direction is equal to the applied force minus the frictional force.
Net force in the horizontal direction = Fx - Ff = 155.25 N - 52.97 N = 102.28 N
Finally, we can find the acceleration of the box using Newton's second law:
Net force = mass * acceleration
102.28 N = 27 kg * acceleration
acceleration = 102.28 N / 27 kg ≈ 3.788 m/s^2
Therefore, the acceleration of the box is approximately 3.788 m/s^2.