A box of books has a mass of 46kg and is being towed by a rope which is oriented at 15 degree to the horizontal at a point where it is tied into a knot. If the pulling force is increased until the box begins to slide, determine the box's initial acceleration if the coefficient of static friction is 0.4 and the coefficient of dynamic friction is 0.25
at the point where it has acceleration = a and velocity = 0, I assume the static friction rules.
normal force = m g - F sin 15
friction force = .4(mg-F sin 15)
pull force horizontal = F cos 15
so
Fcos15 -.4(mg-F sin 15)= m a
-60
To determine the box's initial acceleration, we need to consider the forces acting on the box.
1. Weight (mg): The weight of the box can be calculated using the mass and acceleration due to gravity. Since the box is at rest initially, the weight force acts vertically downwards and can be calculated as follows:
Weight (mg) = mass * acceleration due to gravity
Weight = 46 kg * 9.8 m/s^2 = 450.8 N
2. Normal force (N): The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, the normal force acts perpendicular to the inclined plane. It can be calculated using the following formula:
Normal force (N) = Weight * cos(θ)
N = 450.8 N * cos(15°) ≈ 429.8 N
Note: The angle θ is the angle between the inclined plane and the horizontal. In this case, θ = 15°.
3. Static friction force (f_s): The static friction force opposes the applied force until the maximum value is reached. This maximum value is given by the coefficient of static friction multiplied by the normal force:
Static friction force (f_s) = coefficient of static friction * Normal force
f_s = 0.4 * 429.8 N ≈ 171.9 N
4. Applied force (F): The applied force is the force exerted by towing, and it is directed parallel to the inclined plane. We need to find the minimum applied force required to overcome static friction and initiate motion:
Minimum Applied force (F_min) = Static friction force (f_s)
F_min = 171.9 N
5. Net force (F_net): The net force acting on the box is the difference between the applied force and the friction force:
Net force (F_net) = Applied force (F) - Friction force (f_s)
Since the box is at rest initially, F = F_min
F_net = F_min - f_s
F_net = 171.9 N - 171.9 N = 0 N
Since the net force is zero, there is no acceleration. The box will remain at rest until a force greater than the static friction force is applied.
Therefore, the box's initial acceleration is 0 m/s^2.