A 5.1–kg block is pulled along a frictionless floor by a cord that exerts a force P = 12 N at an angle θ = 25° above the horizontal, as shown below. (a) What is the acceleration of the
block? (b) The force P is slowly increased. What is the value of P just before the block is lifted off the floor? (c) Determine the acceleration of the block just before it is lifted off the floor?
a) The acceleration of the block is 2.4 m/s^2.
b) The value of P just before the block is lifted off the floor is 24 N.
c) The acceleration of the block just before it is lifted off the floor is 4.8 m/s^2.
To solve this problem, we can break down the force P into its horizontal and vertical components.
(a) The acceleration of the block can be found using Newton's second law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
Since the floor is frictionless, the only horizontal force acting on the block is the horizontal component of the force P. This component is given by P_horizontal = P * cos(θ).
So, P_horizontal = 12 N * cos(25°) = 10.90 N.
The vertical component of the force P does not contribute to the horizontal motion of the block. Therefore, we can ignore it for now.
Using Newton's second law, we can say that P_horizontal = m * a, where m is the mass of the block and a is the acceleration. Rearranging the equation, we have:
a = P_horizontal / m = 10.90 N / 5.1 kg = 2.14 m/s^2.
Therefore, the acceleration of the block is 2.14 m/s^2.
(b) To find the value of P just before the block is lifted off the floor, we need to consider the vertical component of the force P. This vertical component opposes the gravitational force acting on the block.
The weight of the block can be calculated using the formula:
Weight = mass * acceleration due to gravity = m * g,
where g is the acceleration due to gravity, approximately 9.8 m/s^2.
Weight = 5.1 kg * 9.8 m/s^2 = 50.28 N.
The vertical component of the force P is equal in magnitude but opposite in direction to the weight of the block. So, P_vertical = weight = 50.28 N.
To find the value of P just before the block is lifted off the floor, we need to find the minimum horizontal force that can balance the vertical component of the force P, i.e., P_vertical.
P_horizontal = P * cos(θ) = P_vertical.
Therefore,
P * cos(θ) = 50.28 N.
Solving for P, we have:
P = 50.28 N / cos(θ) = 50.28 N / cos(25°) ≈ 54.29 N.
So, the value of P just before the block is lifted off the floor is approximately 54.29 N.
(c) To determine the acceleration of the block just before it is lifted off the floor, we need to consider the total force acting on the block at that moment. When the block is just about to be lifted off the floor, the force P is equal to the maximum horizontal force required to overcome the static friction between the block and the floor.
The maximum static friction force is given by the equation:
Static friction force = coefficient of static friction * normal force.
Since the floor is frictionless, the normal force acting on the block is equal to its weight, which we found to be 50.28 N.
Using the equation above, we have:
Static friction force = 0 * 50.28 N = 0 N.
Therefore, the maximum static friction force is zero, meaning that there is no friction force acting on the block just before it is lifted off the floor.
In the absence of any friction force, the only horizontal force acting on the block is the horizontal component of the force P (P_horizontal). So, the acceleration of the block just before it is lifted off the floor will be the same as the acceleration we calculated in part (a), which is 2.14 m/s^2.
To find the acceleration of the block, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
(a) The force P can be resolved into its vertical and horizontal components. The vertical component, P_y, is given by P_y = P * sin(θ), and the horizontal component, P_x, is given by P_x = P * cos(θ). In this case, P_y = 12 N * sin(25°) and P_x = 12 N * cos(25°).
Since there is no friction on the floor, the horizontal component of the force does not affect the acceleration of the block. Therefore, the only force acting on the block in the horizontal direction is the force applied by the cord, which is equal to P_x.
Using Newton's second law, the net force in the horizontal direction is given by F_net = m * a, where m is the mass of the block and a is its acceleration. Since there is no other force acting in the horizontal direction, F_net is equal to P_x. Therefore, we have P_x = m * a.
Substituting the values, we get 12 N * cos(25°) = 5.1 kg * a. Solving this equation will give us the value of the acceleration.
(b) To find the value of P just before the block is lifted off the floor, we need to consider the vertical forces acting on the block. The weight of the block, which is equal to the force acting due to gravity, is given by W = m * g, where g is the acceleration due to gravity. In this case, W = 5.1 kg * 9.8 m/s^2.
The vertical component of the force applied by the cord, P_y, has to balance the weight of the block to keep it on the floor. Therefore, we have P_y = W. Substituting the values, we get 12 N * sin(25°) = 5.1 kg * 9.8 m/s^2. Solving this equation will give us the value of P just before the block is lifted off the floor.
(c) To determine the acceleration of the block just before it is lifted off the floor, we need to consider the net force acting on the block in the vertical direction. The net force is given by F_net = P_y - W.
Just before the block is lifted off the floor, the net force should be enough to overcome the weight of the block. Therefore, the net force should be greater than or equal to zero. Setting F_net >= 0 and substituting the values, we can solve for the value of acceleration.