Assume cathy has to drag her suitcase along the floor of the airport terminal by a rope. The rope makes a 40.0 degree angle with the horizantal. The suitcase has a mass of 36.0 kg and Cathy pulls on the rope with a force of 65.0 N.
a) What is the magnitude of the normal force acting on the suitcase due to the floor?
b) If the kinetic friction between the suitcase and the marble floor is 40.0 N, find the acceleration of the suitcase while Cathy pulls with 65.0 N force at 40 degrees.
c) Starting from rest, for how long a time must she pull with this force until the suitcase reaches a comfortable walking speed of 0.5 m/s^2.
To find the solution to these problems, we can apply Newton's laws of motion. Let's break it down step by step:
a) The normal force is the force exerted perpendicular to the surface of contact. In this case, it is the force exerted by the floor on the suitcase in the upward direction to balance the weight of the suitcase.
To find the magnitude of the normal force, we need to resolve the force components in the horizontal and vertical directions. Considering the given information, the vertical component of the force is the weight of the suitcase, which is given by:
Weight = mass * acceleration due to gravity
= 36.0 kg * 9.8 m/s^2
= 352.8 N
Since the suitcase is placed on a flat surface, there is no vertical acceleration, which means the vertical forces are balanced. Therefore, the magnitude of the normal force is equal to the weight of the suitcase:
Normal force = Weight
= 352.8 N
So, the magnitude of the normal force acting on the suitcase due to the floor is 352.8 N.
b) The acceleration of the suitcase can be determined using the following equation:
Net force = mass * acceleration
The net force is the vector sum of the force applied by Cathy and the frictional force. The force applied by Cathy can be separated into horizontal and vertical components. The horizontal component is given by:
Force horizontal = Force applied * cos(angle)
= 65.0 N * cos(40.0 degrees)
≈ 49.75 N
The frictional force acts in the opposite direction to the motion of the suitcase. So, the magnitude of the frictional force is given as 40.0 N. Therefore, the net horizontal force is:
Net force horizontal = Force horizontal - Frictional force
= 49.75 N - 40.0 N
= 9.75 N
Now we can calculate the acceleration using Newton's second law:
Net force horizontal = mass * acceleration
9.75 N = 36.0 kg * acceleration
acceleration ≈ 0.271 m/s^2
So, the acceleration of the suitcase while Cathy pulls with a force of 65.0 N at 40 degrees is approximately 0.271 m/s^2.
c) To find the time taken for the suitcase to reach a comfortable walking speed of 0.5 m/s^2, we can use the following kinematic equation:
vf = vi + at,
where vf is the final velocity, vi is the initial velocity (0 m/s), a is the acceleration (0.5 m/s^2), and t is the time taken.
Rearranging the equation, we get:
t = (vf - vi) / a
= vf / a
Using the acceleration found in part b, we have:
t = 0.5 m/s / 0.271 m/s^2
≈ 1.845 seconds
So, approximately 1.845 seconds is the time Cathy must pull with a force of 65.0 N until the suitcase reaches a comfortable walking speed of 0.5 m/s^2.