A horizontal force F is exerted on a 20-kg box to slide it up a 30 degrees incline. The friction force ing the motion is 80 N.How large must F be if the acceleration of the moving box is to (a)zero and (b) 0.75m/s2?
To find the required force for the box to have a zero acceleration or a specific acceleration, we need to consider the forces acting on the box and apply Newton's second law of motion, which states that the net force on an object is equal to the mass of the object multiplied by its acceleration.
Let's break down the forces acting on the box:
1. Weight: The weight of the box can be calculated using the formula W = m * g, where W is the weight, m is the mass of the box, and g is the acceleration due to gravity (approximately 9.8 m/s²). Therefore, the weight of the box is W = 20 kg * 9.8 m/s² = 196 N.
2. Normal Force: The normal force (N) is the perpendicular force exerted by a surface to support the weight of an object resting on it. In this case, the normal force is equal to the component of the weight perpendicular to the incline, which can be calculated using the formula N = W * cos(theta), where theta is the angle of the incline (30 degrees). Therefore, N = 196 N * cos(30 degrees) = 169.91 N.
3. Friction Force: The friction force (Ff) opposes the motion of the box and can be calculated using the formula Ff = u * N, where u is the coefficient of friction and N is the normal force. In this case, the friction force is given as 80 N.
Now, let's calculate the required force (F) for each scenario:
(a) For zero acceleration, the net force on the box must be zero. This means that the force exerted by us (F) must be equal to the sum of the friction force and the component of the weight parallel to the incline. The component of the weight parallel to the incline can be calculated using the formula W * sin(theta), where theta is the angle of the incline (30 degrees). Therefore, F = Ff + W * sin(theta).
F = 80 N + 196 N * sin(30 degrees) = 80 N + 98 N = 178 N.
So, the force required for the box to have zero acceleration is 178 N.
(b) For an acceleration of 0.75 m/s², the net force on the box must be equal to the mass of the box multiplied by the desired acceleration. Therefore, the force required can be calculated using the formula F = m * a.
F = 20 kg * 0.75 m/s² = 15 N.
So, the force required for the box to have an acceleration of 0.75 m/s² is 15 N.