A 4.1 kg block initially at rest is pulled to the right along a horizontal surface by a constant, horizontal force of 12.6 N. The coefficient of kinetic friction is 0.103.
The acceleration of gravity is 9.8 m/s^2.
Find the speed of the block after it has moved 2.12 m.
Answer in units of m/s.
weight = mg = 4.1 * 9.8 = 40.2 Newtons
Horizontal friction = 0.103 * 40.2 = 4.14 Newtons
net horizontal force = 12.6 - 4.14 = 8.46 N
a = F/m = 8.46 / 4.1 = 2.06 m/s^2
v = a t
x = (1/2) a t^2
2.12 = 1.03 t^2
t = 1.435 seconds
v = 2.06 m/s^2 * 1.435 s = 2.96 m/s
Why did the block go to the therapist?
Because it needed help with its friction issues!
But don't worry, I've got your back! Let's solve this together.
To find the speed of the block, we first need to calculate the net force acting on it. We start by calculating the force of friction:
Friction = coefficient of kinetic friction x normal force
The normal force is equal to the weight of the block, which is:
Normal force = mass x gravity
Now, the force pulling the block to the right is 12.6 N. Since there is no vertical motion, this is equal to the normal force:
Normal force = 12.6 N
Plugging in the given data, we get:
Friction = 0.103 x (4.1 kg x 9.8 m/s^2)
Friction = 4.1 N
Since the block is pulled to the right, the friction acts to the left, opposing the motion.
Now let's calculate the net force:
Net force = pulling force - friction
Net force = 12.6 N - 4.1 N
Net force = 8.5 N
We can now use Newton's second law to find the acceleration:
Net force = mass x acceleration
8.5 N = 4.1 kg x acceleration
acceleration = 8.5 N / 4.1 kg
acceleration ≈ 2.073 m/s^2
Now, we can use the equation of motion to find the speed (v) of the block after it has moved 2.12 m:
v^2 = u^2 + 2as
Given that the initial velocity (u) is 0 (since the block was initially at rest), and the acceleration (a) is 2.073 m/s^2, we can solve for v:
v^2 = 0^2 + 2 x 2.073 m/s^2 x 2.12 m
v^2 = 8.79976 m^2/s^2
v ≈ 2.97 m/s
So, the speed of the block after it has moved 2.12 m is approximately 2.97 m/s.
To find the speed of the block after it has moved 2.12 m, we can use the equations of motion along with Newton's second law and the concepts of friction.
1. Determine the net force acting on the block:
The force acting on the block is the applied force minus the frictional force:
Net force = Applied force - Frictional force
The applied force is given as 12.6 N.
The frictional force can be calculated using the formula:
Frictional force = 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:
Normal force = mass * gravity
The mass of the block is given as 4.1 kg, and the acceleration due to gravity is given as 9.8 m/s^2.
Plugging in the values, we can calculate the net force:
Frictional force = 0.103 * (4.1 kg * 9.8 m/s^2)
Frictional force ≈ 4.045 N
Net force = 12.6 N - 4.045 N
Net force ≈ 8.555 N
2. Calculate the acceleration of the block:
Using Newton's second law, we know that the net force is equal to the mass of the object multiplied by its acceleration:
Net force = mass * acceleration
Plugging in the values, we can solve for the acceleration:
8.555 N = 4.1 kg * acceleration
acceleration ≈ 2.085 m/s^2
3. Calculate the final velocity of the block:
Using the equations of motion, we can find the final velocity of the block.
We know the initial velocity is zero because the block starts from rest, and we know the distance traveled is 2.12 m.
The equation relates initial velocity (u), final velocity (v), acceleration (a), and distance (s) is:
v^2 = u^2 + 2as
Since the initial velocity is zero, we can simplify the equation to:
v^2 = 2as
Plugging in the values, we can solve for the final velocity:
v^2 = 2 * 2.085 m/s^2 * 2.12 m
v^2 ≈ 8.846 m^2/s^2
Taking the square root of both sides, we find:
v ≈ √8.846 m^2/s^2
v ≈ 2.977 m/s
Therefore, the speed of the block after it has moved 2.12 m is approximately 2.977 m/s.
To find the speed of the block after it has moved 2.12 m, we need to analyze the forces acting on the block and apply Newton's second law to determine its acceleration.
The forces acting on the block are the applied force pulling it to the right and the force of kinetic friction opposing its motion. The equation for the magnitude of the force of kinetic friction is given by:
Force of kinetic friction = coefficient of kinetic friction * normal force
The normal force acting on the block is equal to its weight, which is given by:
Normal force = mass * acceleration due to gravity
Therefore, the force of kinetic friction can be calculated as:
Force of kinetic friction = coefficient of kinetic friction * mass * acceleration due to gravity
Now, using Newton's second law (F = ma), we can write an equation for the net force acting on the block:
Net force = force applied - force of kinetic friction
Since the acceleration of the block is constant, we can solve for its acceleration:
Acceleration = net force / mass
Next, we can use the kinematic equation to find the final speed of the block:
V^2 = V_0^2 + 2as
Where V is the final speed, V_0 is the initial speed (0 m/s since the block starts at rest), a is the acceleration, and s is the distance traveled (2.12 m in this case).
Let's calculate the speed step by step:
Step 1: Calculate the force of kinetic friction.
Force of kinetic friction = 0.103 * 4.1 kg * 9.8 m/s^2
Step 2: Calculate the net force.
Net force = 12.6 N - Force of kinetic friction (from Step 1)
Step 3: Calculate the acceleration.
Acceleration = Net force / mass
Step 4: Calculate the final speed.
V^2 = 0 + 2 * acceleration * 2.12 m
Finally, solve for V by taking the square root of both sides of the equation and convert the final answer to the correct units of m/s.