A block is pushed across a horizontal surface with a coefficient of kinetic friction of 0.15 by applying a 150 N horizontal force.
a) The block accelerates at the rate of 2.53 m/s^2. Find the mass of the block.
b) The block slides across a new surface while experiencing the same applied force as before. The block now moves with a constant speed. What is the coefficient of kinetic friction between the block and the new surface?
a) To find the mass of the block, we can use the equation for force and acceleration:
Force = mass * acceleration
The force in this case is the applied force of 150 N, and the acceleration is 2.53 m/s^2. Plugging these values into the equation:
150 N = mass * 2.53 m/s^2
Rearranging the equation to solve for mass:
mass = 150 N / 2.53 m/s^2
mass ≈ 59.28 kg
Therefore, the mass of the block is approximately 59.28 kg.
b) When the block is sliding across the new surface at a constant speed, the applied force is balanced by the force of kinetic friction. The equation for this is:
Force of kinetic friction = coefficient of kinetic friction * normal force
Since the normal force is equal to the weight of the block (mg), and the applied force is 150 N, we can rewrite the equation as:
150 N = coefficient of kinetic friction * mg
We know from part a) that the mass of the block is 59.28 kg. The acceleration due to gravity is approximately 9.8 m/s^2. Plugging in these values:
150 N = coefficient of kinetic friction * 59.28 kg * 9.8 m/s^2
Simplifying the equation:
coefficient of kinetic friction = 150 N / (59.28 kg * 9.8 m/s^2)
coefficient of kinetic friction ≈ 0.267
Therefore, the coefficient of kinetic friction between the block and the new surface is approximately 0.267.
To find the mass of the block in part a), we need to use Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. The formula is:
Force = mass * acceleration
In this case, the force exerted is 150 N and the acceleration is 2.53 m/s^2. We can rearrange the formula to solve for mass:
mass = Force / acceleration
Plugging in the given values:
mass = 150 N / 2.53 m/s^2 = 59.29 kg
Therefore, the mass of the block is 59.29 kg.
To find the coefficient of kinetic friction between the block and the new surface in part b), we can use the formula:
force of friction = coefficient of friction * normal force
In this case, the applied force is still 150 N and the block is moving with a constant speed, which means the acceleration is zero. When the block moves at a constant speed, the force of friction is equal to the applied force.
So, force of friction = 150 N.
We can rearrange the formula to solve for the coefficient of friction:
coefficient of friction = force of friction / normal force
Since the block is not accelerating vertically, the normal force is equal to the weight of the block, which is given by:
normal force = mass * gravity
Assuming the acceleration due to gravity is 9.8 m/s^2, we can calculate the normal force:
normal force = 59.29 kg * 9.8 m/s^2 = 581.14 N
Now, we can calculate the coefficient of kinetic friction:
coefficient of friction = 150 N / 581.14 N = 0.258
Therefore, the coefficient of kinetic friction between the block and the new surface is 0.258.
a) To find the mass of the block, we can use Newton's second law of motion: F = ma. In this case, the force is the applied force minus the force of friction.
The applied force is given as 150 N, and the coefficient of kinetic friction is given as 0.15. The force of friction can be calculated using the formula F_friction = coefficient of kinetic friction * normal force.
Since the block is on a horizontal surface, the normal force is equal to the weight of the block, which is given by the formula F_gravity = mg, where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Now we can set up the equation:
150 N - F_friction = ma
150 N - (0.15 * mg) = ma
150 N - (0.15 * m * 9.8 m/s^2) = m * 2.53 m/s^2
Simplifying the equation:
150 N - 1.47 m = 2.53 m^2/s^2
Rearranging the equation to solve for mass:
1.47 m = 150 N - 2.53 m^2/s^2
1.47 m + 2.53 m^2/s^2 = 150 N
m(1.47 + 2.53 m/s^2) = 150 N
m = 150 N / (1.47 + 2.53 m/s^2)
Using a numerical approach or solving the quadratic equation, we find that the mass of the block is approximately 19.88 kg.
b) Since the block is moving with a constant speed, the net force acting on it must be zero. The force of friction is equal to the applied force in this case.
Using the same formula as before:
F_friction = coefficient of kinetic friction * normal force
150 N = coefficient of kinetic friction * (m * g)
Dividing both sides by (m * g):
150 N / (m * g) = coefficient of kinetic friction
Substituting the mass we found in part a (19.88 kg) and the acceleration due to gravity (9.8 m/s^2):
150 N / (19.88 kg * 9.8 m/s^2) ≈ 0.774
Therefore, the coefficient of kinetic friction between the block and the new surface is approximately 0.774.