A 30.0-kg block is resting on a flat horizontal table. On top of this block is resting a 15.0-kg block, to which a horizontal spring is attached, as the drawing illustrates. The spring constant of the spring is 325 N/m. The coefficient of kinetic friction between the lower block and the table is 0.600, and the coefficient of static friction between the two blocks is 0.900. A horizontal force F is applied to the lower block as shown. This force is increasing in such a way as to keep the blocks moving at a constant speed. At the point where the upper block begins to slip on the lower block, determine (a) the amount by which the spring is compressed and (b) the magnitude of the force
F.
a) The amount by which the spring is compressed is 0.9 m.
b) The magnitude of the force F is 45.0 N.
To determine the amount by which the spring is compressed and the magnitude of the force, we can follow these steps:
Step 1: Calculate the force of static friction between the two blocks.
The force of static friction can be calculated using the equation: f_static = μ_static * N, where μ_static is the coefficient of static friction and N is the normal force.
The normal force can be calculated using the equation: N = mass * acceleration due to gravity.
For the 15.0 kg block, N = 15.0 kg * 9.8 m/s² = 147.0 N.
The force of static friction is: f_static = 0.900 * 147.0 N = 132.3 N.
Step 2: Calculate the force of kinetic friction between the lower block and the table.
The force of kinetic friction can be calculated using the equation: f_kinetic = μ_kinetic * N, where μ_kinetic is the coefficient of kinetic friction and N is the normal force.
For the 30.0 kg block, N = 30.0 kg * 9.8 m/s² = 294.0 N.
The force of kinetic friction is: f_kinetic = 0.600 * 294.0 N = 176.4 N.
Step 3: Calculate the net force acting on the blocks.
Since the blocks are moving at a constant speed, the net force acting on them must be zero.
The net force can be calculated using the equation: net_force = F_applied - f_static - f_kinetic.
Substituting the given values, we have:
0 = F_applied - 132.3 N - 176.4 N.
Simplifying the equation, we get: F_applied = 308.7 N.
Step 4: Calculate the displacement of the spring (amount by which it is compressed).
The force required to compress or extend a spring is given by Hooke's Law: F_spring = -k * x, where F_spring is the force, k is the spring constant, and x is the displacement of the spring.
In this case, the force of the spring is equal to the applied force: F_spring = F_applied = 308.7 N.
The spring constant is given as 325 N/m.
Substituting these values, we have: 308.7 N = -325 N/m * x.
Solving for x, we get: x = -308.7 N / (-325 N/m) = 0.95 m (rounded to two decimal places).
Therefore, the spring is compressed by 0.95 meters.
To summarize:
(a) The amount by which the spring is compressed is 0.95 meters.
(b) The magnitude of the force applied (F_applied) is 308.7 Newtons.
To determine the amount by which the spring is compressed and the magnitude of the force, let's break down the problem step by step.
Step 1: Analyze the forces acting on the system.
Considering the lower block and applying Newton's second law (F = ma), we have:
F_net = F_applied - F_friction
Where:
- F_net is the net force acting on the lower block,
- F_applied is the applied force,
- F_friction is the frictional force.
Step 2: Calculate the frictional force on the lower block.
The coefficient of kinetic friction (μk) between the lower block and the table is given as 0.600. The frictional force (F_friction) can be calculated as:
F_friction = μk * (normal force)
The normal force (F_normal) is the force exerted by the table on the lower block, which is equal to the weight of the lower block (mg).
F_normal = m * g
F_friction = μk * (m * g)
Step 3: Determine the net force on the lower block.
The net force can now be calculated by substituting the values:
F_net = F_applied - F_friction
F_net = F_applied - (μk * (m * g))
Step 4: Calculate the acceleration of the system.
Since the blocks are moving at a constant speed, the net force must be zero. Therefore:
F_net = 0
F_applied - (μk * (m * g)) = 0
F_applied = μk * (m * g)
Step 5: Find the spring compression.
The force applied to the lower block (F_applied) is transmitted to the upper block via the spring. The spring obeys Hooke's Law, which states:
F_spring = -k * x
Where:
- F_spring is the force exerted by the spring,
- k is the spring constant,
- x is the compression or displacement of the spring from its equilibrium position.
The force exerted by the spring (F_spring) can be equated to the applied force (F_applied):
F_spring = F_applied
Substituting the values:
-k * x = μk * (m * g)
Solving for x:
x = (μk * (m * g)) / -k
Step 6: Calculate the force of the spring on the upper block.
The magnitude of the force exerted by the spring on the upper block can be calculated using Hooke's Law:
F_spring = -k * x
Substituting the values:
F_spring = -k * [(μk * (m * g)) / -k]
F_spring = μk * (m * g)
Substituting the given values:
F_spring = 0.600 * (15.0 kg * 9.8 m/s²)
Finally, calculate the values:
(a) The amount by which the spring is compressed is x = (0.600 * (15.0 kg * 9.8 m/s²)) / -325 N/m.
(b) The magnitude of the force exerted by the spring is F_spring = 0.600 * (15.0 kg * 9.8 m/s²).