While trying to decide where to hang a framed picture, you press it against the wall to keep it from falling. The picture weights 9.5 N and you press against the frame with a force of 19 N at an angle of 30° from the vertical.
What is the coefficient of static friction between the wall and the picture?
The frictional force exerted on the picture by the wall can have two possible directions. Explain why.
To determine the coefficient of static friction between the wall and the picture, we can analyze the forces acting on the picture when it is pressed against the wall to keep it from falling.
First, let's resolve the force applied on the picture into its vertical and horizontal components. The vertical component is given by F_vert = 19 N * sin(30°) = 9.5 N, which balances the weight (9.5 N) of the picture.
Since the picture is not falling or sliding down, this means that the static friction force between the wall and the picture is equal in magnitude and opposite in direction to the horizontal component of the pressing force.
From the given information, the horizontal component of the pressing force is F_horiz = 19 N * cos(30°) = 16.5 N.
The maximum static friction force (F_max) can be given by:
F_max = coefficient of static friction * Normal force
In this case, the normal force is equal to the weight of the picture, which is 9.5 N.
Using this information, we can set up the equation:
F_max = coefficient of static friction * 9.5 N
Since the maximum static friction force is also equal to the magnitude of the horizontal pressing force (F_horiz), we can rewrite the equation as:
F_horiz = coefficient of static friction * 9.5 N
Plugging in the values we found earlier, we get:
16.5 N = coefficient of static friction * 9.5 N
Simplifying, we find:
coefficient of static friction = 16.5 N / 9.5 N
coefficient of static friction ≈ 1.74
Now, let's discuss why the frictional force exerted on the picture by the wall can have two possible directions.
Since the picture is being pressed against the wall, the wall exerts a normal force perpendicular to its surface. The static friction force acts parallel to the surface of contact between the picture and the wall.
The direction of the frictional force depends on the direction of the applied force. In this case, the applied force is angled at 30° from the vertical, so the frictional force can act in two opposite directions: either towards the left or towards the right.
The direction of the frictional force will be such that it opposes the horizontal component of the applied force. If the applied force is towards the left, the frictional force will be towards the right, and vice versa. This ensures that the picture does not slide horizontally and remains in equilibrium against the wall.
To determine the coefficient of static friction between the wall and the picture, we first need to understand the forces at play.
When you press the picture against the wall, there are two forces acting on the picture:
1. The weight of the picture, which is a downward force of 9.5 N.
2. The force you exert on the picture, which is a force of 19 N at an angle of 30° from the vertical.
Now, let's analyze the direction of the frictional force exerted on the picture by the wall:
1. If the coefficient of static friction is sufficient, the wall will exert an upward frictional force on the picture, opposing the picture's downward weight. This would keep the picture in place against the wall.
2. If the coefficient of static friction is not strong enough, the picture would start to slide down the wall. In this case, the frictional force would act in the opposite direction, opposing the upwards force you exert on the picture, causing it to slide.
In other words, the direction of the frictional force can either be upward or downward depending on whether the coefficient of static friction is strong enough to prevent the picture from sliding or not.
To calculate the coefficient of static friction, we can set up the equation of equilibrium for the vertical direction:
Sum of vertical forces = 0
The forces in the vertical direction are:
- The downward weight of the picture (9.5 N)
- The upward frictional force exerted by the wall (unknown, let's call it F_friction)
Therefore, the equation becomes:
9.5 N - F_friction = 0
Solving for F_friction gives us:
F_friction = 9.5 N
However, since we don't yet know the coefficient of static friction, we can't directly calculate the value of F_friction. We need additional information to determine the coefficient of static friction.
To calculate the coefficient of static friction, we need to know the normal force exerted by the wall on the picture. This normal force is equal to the weight of the picture when there is no vertical acceleration:
Normal force = weight of the picture = 9.5 N
Now we can use the formula for static friction:
F_friction = coefficient of static friction × normal force
Plugging in the values we have:
9.5 N = coefficient of static friction × 9.5 N
Dividing both sides by 9.5 N provides the coefficient of static friction:
Coefficient of static friction = 1
Therefore, the coefficient of static friction between the wall and the picture is 1.