A block of mass m1 = 4.90 kg sits on top of a second block of mass m2 = 14.5 kg, which in turn is on a horizontal table. The coefficients of friction between the two blocks are µs = 0.300 and µk = 0.100. The coefficients of friction between the lower block and the rough table are µs = 0.500 and µk = 0.400. You apply a constant horizontal force to the lower block, just large enough to make this block start sliding out from between the upper block and the table.
(a) Determine the magnitude of each force on each block at the instant when you have started pushing but motion has not yet started. In particular, what force must you apply?
(b) Determine the acceleration you measure for each block.
To solve this problem, we will consider the forces acting on each block separately. Let's start with the lower block.
(a) Forces on the lower block:
There are two forces acting on the lower block: the force of friction between the lower block and the table, and the force applied by you.
1. Force of friction (static) between the lower block and the table (Fsf):
The maximum static friction force between two surfaces can be calculated using the formula:
Fsf = µs * N
where µs is the coefficient of static friction and N is the normal force. The normal force on the lower block is equal to its weight since it is on a horizontal table, so we can calculate it as:
N = m2 * g
where m2 is the mass of the lower block and g is the acceleration due to gravity.
2. Force applied by you (Fapplied):
Since the lower block is on the verge of sliding, the force applied by you must be equal to the force of friction (static) between the lower block and the table:
Fapplied = Fsf
Now let's move on to the upper block.
Forces acting on the upper block:
There are three forces acting on the upper block: the gravitational force, the normal force, and the force of friction between the two blocks.
1. Gravitational force (Fg):
The gravitational force on the upper block is given by:
Fg = m1 * g
2. Normal force (N'):
The normal force on the upper block is equal to the force of gravity (Fg) since it is pressing down on the lower block.
3. Force of friction between the two blocks (Ffk):
The maximum kinetic friction force between the two blocks can be calculated using the formula:
Ffk = µk * N'
where µk is the coefficient of kinetic friction, and N' is the normal force.
Since the lower block is not yet in motion, the force of friction between the two blocks is at its maximum static friction value, which is greater than the applied force. Therefore, the upper block experiences static friction and remains stationary under the force applied by you.
(b) Acceleration of each block:
Since the upper block remains stationary, its acceleration is 0 m/s^2. The acceleration of the lower block can be calculated using Newton's second law:
∑F = m2 * a
where ∑F is the net force acting on the lower block and a is the acceleration of the lower block.
To find the net force on the lower block, we need to consider the forces acting on it:
∑F = Fapplied - Fsf
Now, let's calculate the values step-by-step:
Step 1: Calculate the normal force on the lower block
N = m2 * g
Step 2: Calculate the force of static friction between the lower block and the table
Fsf = µs * N
Step 3: Calculate the force applied by you
Fapplied = Fsf
Step 4: Calculate the normal force on the upper block
N' = Fg
Step 5: Calculate the force of kinetic friction between the two blocks
Ffk = µk * N'
Step 6: Calculate the net force on the lower block
∑F = Fapplied - Fsf
Step 7: Calculate the acceleration of the lower block
a = ∑F / m2
Now, you can substitute the given values (masses, coefficients of friction, and acceleration due to gravity) into the above formulas to find the magnitude of each force on each block and the acceleration of each block.
To determine the magnitude of each force on each block and the force that must be applied to the lower block in order to start sliding, you need to analyze the forces acting on each block.
(a) Forces on the top block:
1. Weight: The weight of the top block is given by W1 = m1 * g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. Normal force: The normal force exerted on the top block by the bottom block is equal in magnitude and opposite in direction to the weight of the top block, N1 = -W1.
Forces on the bottom block (before motion starts):
1. Weight: The weight of the bottom block is given by W2 = m2 * g.
2. Normal force from the table: The normal force exerted on the bottom block by the table is equal in magnitude and opposite in direction to the weight of the bottom block, N2 = -W2.
3. Friction force from the top block: The maximum static friction force that can be exerted by the top block on the bottom block is given by μs * N2, where μs is the coefficient of static friction between the two blocks.
4. Friction force from the table: The maximum static friction force that can be exerted by the table on the bottom block is given by μs * N2, where μs is the coefficient of static friction between the bottom block and the table.
To just start sliding, the applied force must overcome the maximum static friction between the top block and the bottom block. Therefore, the force applied to the lower block must be equal to the maximum static friction force between the two blocks, F_applied = μs * N2.
(b) Once motion starts, the forces change. The friction between the top and bottom blocks transitions from static friction to kinetic friction. The maximum static friction is given by μs * N2, while the kinetic friction is given by μk * N2, where μk is the coefficient of kinetic friction between the two blocks.
The acceleration of each block can be determined using Newton's second law, F = m * a, where F is the net force acting on the block, m is the mass of the block, and a is the acceleration.
For the top block:
Net force = F_applied - kinetic friction force = F_applied - μk * N2
Acceleration = a1 = (F_applied - μk * N2) / m1
For the bottom block:
Net force = μk * N2 - kinetic friction force from the table = μk * N2 - μk * N2 = 0 (since the kinetic friction from the table cancels out with the kinetic friction from the top block)
Acceleration = a2 = 0
Therefore, the acceleration of the top block, a1, will be equal to the force applied to the bottom block divided by the top block's mass, while the acceleration of the bottom block, a2, will be zero.
By substituting the given values for the masses, coefficients of friction, and acceleration due to gravity into the equations, you can calculate the magnitudes of the forces on each block and the applied force required to start sliding.