A block is pressed against a vertical wall by a force , as the drawing shows. This force can either push the block upward at a constant velocity or allow it to slide downward at a constant velocity. The magnitude of the force is different in the two cases, while the directional angle θ is the same. Kinetic friction exists between the block and the wall, and the coefficient of kinetic friction is 0.310. The weight of the block is 52.0 N, and the directional angle for the force is θ = 38.0°. Determine the magnitude of when the block slides (a) up the wall and (b) down the wall.
(a) P =
To determine the magnitude of the force P when the block slides up the wall (against gravity), we can use Newton's second law of motion.
The force diagram for the block when it is moving up the wall is as follows:
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| θ |
| |
| |
| P |
| <--- |
| Block |
| |
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The forces acting on the block when it moves up the wall are:
1. The force P (directed upwards).
2. The force of gravity (52.0 N) acting downwards.
3. The kinetic friction force (opposes the motion and acts downwards).
Since the block is moving up the wall at a constant velocity (constant speed), the net force acting on it is zero. This means that the force due to gravity and the kinetic friction force are balanced by the force P:
P - 52.0 N - (0.310 * normal force) = 0
To determine the normal force, we need to analyze the forces acting on the block in the vertical direction:
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| θ |
| |
| |
| F1 |
| <--- F2 |
| Block |
| F3 |
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The forces acting on the block in the vertical direction are:
1. The force F1, component of the weight of the block acting in the vertical direction.
2. The force F2, component of the force P acting in the vertical direction.
3. The force F3, kinetic friction force acting in the vertical direction.
From the force diagram, we can see that F1 = 52.0 N * cos(θ) and F2 = P * sin(θ).
The sum of the forces in the vertical direction is zero since the block is not accelerating vertically:
F1 + F2 - F3 = 0
Substituting the values of F1 and F2, we get:
(52.0 N * cos(θ)) + (P * sin(θ)) - (0.310 * normal force) = 0
Now, we need to express the normal force in terms of the weight of the block:
normal force = weight of the block (since the block is not moving vertically)
Substituting this into the equation, we have:
(52.0 N * cos(θ)) + (P * sin(θ)) - (0.310 * 52.0 N) = 0
Simplifying the equation, we can solve for P:
P * sin(θ) = (0.310 * 52.0 N) - (52.0 N * cos(θ))
P = ((0.310 * 52.0 N) - (52.0 N * cos(θ))) / sin(θ)
Plugging in the given values:
θ = 38.0°
P = ((0.310 * 52.0 N) - (52.0 N * cos(38.0°))) / sin(38°)
Calculating this expression will give you the magnitude of P when the block slides up the wall.
To determine the magnitude of force P when the block slides up the wall, we first need to analyze the forces acting on the block in that scenario.
When the block is sliding up the wall at a constant velocity, the force of gravity (weight) and the force of kinetic friction are acting on the block.
We can start by resolving the weight force into its components. The weight force can be split into two perpendicular components: one parallel to the wall and one perpendicular to the wall.
The component of weight parallel to the wall (P_parallel) is given by P_parallel = weight * sin(theta), where theta is the angle between the force and the direction perpendicular to the wall. In this case, theta is given as 38.0°.
P_parallel = 52.0 N * sin(38.0°)
P_parallel = 52.0 N * 0.61566
P_parallel = 32.03 N
Next, we need to consider the force of kinetic friction (F_friction) acting on the block. The magnitude of the frictional force can be calculated using the equation F_friction = coefficient of kinetic friction * normal force.
The normal force acting on the block is equal to the perpendicular component of the weight force (P_perpendicular). P_perpendicular = weight * cos(theta).
P_perpendicular = 52.0 N * cos(38.0°)
P_perpendicular = 52.0 N * 0.78801
P_perpendicular = 40.97 N
Then, we can calculate the magnitude of the frictional force:
F_friction = coefficient of kinetic friction * P_perpendicular
F_friction = 0.310 * 40.97 N
F_friction = 12.70 N
Now, since the block is sliding up the wall at a constant velocity, the force pushing it upwards (P) must be equal in magnitude to the combined downward forces (P_parallel + F_friction). Therefore, we can calculate the magnitude of force P as:
P = P_parallel + F_friction
P = 32.03 N + 12.70 N
P = 44.73 N
Therefore, the magnitude of force P when the block slides up the wall is 44.73 N.
(b) To find the magnitude of force P when the block slides down the wall, we can use a similar analysis.
When the block is sliding down the wall at a constant velocity, the forces acting on the block are the force of gravity (weight) and the force of kinetic friction, but in opposite directions.
Following the same steps as in part (a), we have the same values for the weight, coefficient of kinetic friction, and theta.
P_parallel = 32.03 N (from part a)
F_friction = 12.70 N (from part a)
Since the block is sliding down at a constant velocity, the magnitude of the force P pushing it downwards must be equal to the combined upward forces (P_parallel + F_friction). Therefore, we can calculate the magnitude of force P as:
P = P_parallel + F_friction
P = 32.03 N + 12.70 N
P = 44.73 N
Therefore, the magnitude of force P when the block slides down the wall is also 44.73 N.