A 90Kg box is pushed by a horizontal force F at constant speed up a ramp incined at 28 degrees. Determine the magnitude of applied force when the ramp is frictionless and when the coefficient of kinetic friction is 0.18
x-axis - along the ramp, y-axis - perpendicular to the ramp (upwards).
x: F =F(fr) + m•g•sinα,
y: N=mgcos α,
F=μ•N + m•g•sinα = μ• m•g•cos α + m•g•sinα =m•g(μ• cos α+ sinα)
To determine the magnitude of the applied force when the ramp is frictionless, we need to understand the forces acting on the box in this scenario.
When the ramp is frictionless, there are two main forces acting on the box: the force of gravity and the applied force. The force of gravity is given by the formula:
F_gravity = m * g
Where m is the mass of the box (90 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
F_gravity = 90 kg * 9.8 m/s^2 = 882 N
Since the box is moving at a constant speed on the frictionless ramp, the applied force must be equal in magnitude and opposite in direction to the force of gravity. Therefore, the magnitude of the applied force is also 882 N.
Now, let's determine the magnitude of the applied force when the coefficient of kinetic friction is 0.18.
When the ramp has a coefficient of kinetic friction, the frictional force becomes a factor in the calculation.
The frictional force can be determined using the formula:
F_friction = μ * N
Where μ is the coefficient of kinetic friction and N is the normal force. The normal force is the component of the force of gravity that is perpendicular to the ramp's surface, and its value can be found using trigonometry.
N = m * g * cos(θ)
Where θ is the angle of inclination of the ramp (28 degrees).
N = 90 kg * 9.8 m/s^2 * cos(28 degrees) = 769.22 N
Now, we can calculate the frictional force:
F_friction = 0.18 * 769.22 N = 138.46 N
Since the box is moving at a constant speed, the applied force must be equal in magnitude to the frictional force but in the opposite direction. Therefore, the magnitude of the applied force is also 138.46 N.
To summarize:
- When the ramp is frictionless, the magnitude of the applied force is 882 N.
- When the coefficient of kinetic friction is 0.18, the magnitude of the applied force is 138.46 N.