Alisha starts pulling (from rest) a box full of books on a horizontal floor. She has tied the rope to the box, the rope makes an angle of 40.0° relative to the floor. She pulls with a force F of magnitude 120N. The total mass of the box is 50.0kg. She pulls the box in a straight line over a distance of 6.00m. There is 70.0N friction force between the floor and the box. Calculate the final speed of the box
F = 120N[40o].
Fk = 70 N.
d = 6.0 m.
F*Cos40 - Fk = M*a.
120*Cos40 - 70 = 50a,
a = 0.439 m/s^2.
V^2 = Vo^2 + 2a*d = 0 + 2*0.439*6 = 5.27,
V = 2.30 m/s.
To calculate the final speed of the box, we need to determine the net force acting on the box and then use Newton's second law of motion to calculate the acceleration. Finally, we can use the equation for final velocity to find the final speed.
Step 1: Resolve Forces
First, let's resolve the force applied by Alisha into horizontal and vertical components. The applied force F can be broken down into two components:
F_horizontal = F * cos(θ)
F_vertical = F * sin(θ)
Where θ is the angle the rope makes with the floor, which in this case is 40.0°.
Step 2: Calculate the Net Force
The net force acting on an object is the vector sum of all the forces acting on it. In this case, the net force can be calculated by subtracting the frictional force from the horizontal component of the applied force:
Net force (F_net) = F_horizontal - frictional force
Since the frictional force is given as 70.0N, we have:
F_net = F_horizontal - 70.0N
Step 3: Find Acceleration
Using Newton's second law of motion, we know that the net force is equal to the mass of the object multiplied by its acceleration:
F_net = m * a
Rearranging the equation, we can solve for acceleration:
a = F_net / m
Step 4: Calculate Final Velocity
We can now calculate the final velocity using the equation:
v^2 = u^2 + 2 * a * d
Where v is the final velocity (which we need to find), u is the initial velocity (0 m/s as the box starts from rest), a is the acceleration, and d is the distance traveled.
Rearranging the equation, we have:
v = sqrt(2 * a * d)
Step 5: Plug Values and Calculate
Let's plug in the given values and calculate the final speed of the box:
F = 120 N (applied force)
θ = 40.0° (angle of the rope)
m = 50.0 kg (mass of the box)
frictional force = 70.0 N
d = 6.00 m (distance traveled)
First, let's find the horizontal component of the force:
F_horizontal = F * cos(40.0°)
Next, let's calculate the net force:
F_net = F_horizontal - frictional force
Now, we can compute the acceleration:
a = F_net / m
Finally, we can find the final velocity by substituting the values into the equation:
v = sqrt(2 * a * d)
Calculating each step will give us the final speed of the box.