A 14.0-kg chair is at rest on a flat floor. The coefficient of static friction between
the chair and the floor is 0.450. A person tries to move the chair by pushing on the
chair with a force directed at an angle of 30„a below the horizontal. What is the
minimum force that the person must apply in order to move the chair?
See previous post.
First off the normal force is m*a so 14kg * 9.8 is 137.2N. Since the person isn't pushing straight not all force goes directly to moving the chair. So 137.2=xcos30. x=158.42N.
To find the minimum force that the person must apply in order to move the chair, we need to calculate the force of static friction.
The force of static friction can be calculated using the formula:
F_friction = coefficient of static friction * normal force
The normal force is the perpendicular force exerted by the floor on the chair, which is equal to the weight of the chair. The weight of the chair can be calculated using the formula:
Weight = mass * acceleration due to gravity
Now, let's calculate the weight of the chair:
Weight = mass * acceleration due to gravity
= 14.0 kg * 9.8 m/s^2
≈ 137.2 N
Next, let's calculate the force of static friction:
F_friction = coefficient of static friction * normal force
= 0.450 * 137.2 N
≈ 61.74 N
Now, the person is pushing on the chair with a force directed at an angle of 30 degrees below the horizontal. We need to resolve this force into horizontal and vertical components.
The horizontal component of the force can be calculated using the formula:
F_horizontal = F_push * cos(angle)
The vertical component of the force can be calculated using the formula:
F_vertical = F_push * sin(angle)
Since we are interested in the minimum force, we need to consider the case where the vertical component of the force counteracts the force of gravity.
The vertical component of the force (F_vertical) must be equal to the weight of the chair (137.2 N) to prevent it from moving vertically. Therefore:
F_vertical = 137.2 N
Now, let's solve for the horizontal component of the force:
F_horizontal = F_push * cos(angle)
= F_push * cos(30 degrees)
To find the minimum force that the person must apply, the horizontal component of the force (F_horizontal) must be equal to or greater than the force of static friction (F_friction). Therefore:
F_horizontal ≥ F_friction
F_push * cos(30 degrees) ≥ 61.74 N
Now, we can solve for the minimum force:
F_push ≥ 61.74 N / cos(30 degrees)
F_push ≥ 71.12 N
Therefore, the minimum force that the person must apply in order to move the chair is approximately 71.12 N.