A spring with a force constant of 100 N/m is used to push a 0.27 kg block of wood against a wall. (a) Find the minimum compression of the spring needed to keep the block from falling, given that the coefficient of static friction between the block and the wall is 0.53.
You need a horizontal force of F = m g/.53 to keep the block from sliding.
F = k x
solve for x
To find the minimum compression of the spring needed to keep the block from falling, we can use the following steps:
Step 1: Identify the forces acting on the block.
In this scenario, there are three forces acting on the block:
- The force due to gravity acting downwards (mg)
- The normal force exerted by the wall acting upwards (N)
- The force due to static friction between the block and the wall (fs)
Step 2: Determine the normal force.
Since the block is pushing against the wall vertically, the normal force is equal to the weight of the block. Therefore, N = mg, where m is the mass of the block and g is the acceleration due to gravity (9.8 m/s^2).
N = (0.27 kg) * (9.8 m/s^2) = 2.646 N
Step 3: Calculate the maximum force due to static friction.
The maximum force due to static friction can be calculated using the formula fsmax = μs * N, where μs is the coefficient of static friction.
fsmax = (0.53) * (2.646 N) = 1.402 N
Step 4: Determine the force exerted by the spring.
The force exerted by the spring is given by Hooke's Law: F = k * x, where k is the force constant of the spring and x is the compression of the spring.
F = k * x = 100 N/m * x
Step 5: Equate the forces to find the minimum compression.
Since the spring force must balance the maximum force due to static friction, we have:
F = fsmax
100 N/m * x = 1.402 N
Step 6: Solve for the minimum compression of the spring.
Divide both sides by 100 N/m to isolate x:
x = 1.402 N / 100 N/m
x = 0.01402 m
Therefore, the minimum compression of the spring needed to keep the block from falling is approximately 0.01402 meters.