A 12 kg wagon is being pulled at an angle of 38degrees above horizontal. What force is applied to
the wagon if it accelerates from rest to a speed of 2.2 m/s over a distance of 3.4 m?
I not sure how to do this question. Is work involved for this question. I am completely stumped.
You can do it with work if you wish or you can do it with F = m a
with work
work done = Force in direction of motion * distance
Force in direction of motion = F cos 38
= .788 F
work done = .788 F * 3.4 meters
= 2.68 F Joules
that is the increase in kinetic energy
(1/2) m v^2 = .5*12 * 2.2^2
= 29 Joules
so
2.68 F = 29
F = 10.8 Newtons
with F = m a
f cos 38 = .788 F again
a = change in velocity/time
= 2.2/time
but time = distance/average speed
= 3.4 /1.1 = 3.09 seconds
so
a = 2.2/3.09 = .712 m/s^2
so
.788 F = 12 * .712
F = 10.8 Newtons (remarkable)
To solve this problem, we need to break it down into steps and use relevant equations from Newton's laws of motion. Work is not directly involved in this specific problem, but some related concepts will be explained.
Step 1: Analyze the given information:
- Mass of the wagon (m): 12 kg
- Angle above horizontal (θ): 38 degrees
- Initial velocity of the wagon (u): 0 m/s
- Final velocity of the wagon (v): 2.2 m/s
- Distance traveled (d): 3.4 m
Step 2: Separate the forces:
- The force applied to the wagon can be divided into two components:
1. Horizontal component (Fx) required to accelerate the wagon.
2. Vertical component (Fy) required to lift the wagon against gravity.
Step 3: Determine the acceleration:
- We can use the kinematic equation v^2 = u^2 + 2as, where a is the acceleration.
- Rearranging the equation, we get a = (v^2 - u^2) / (2d).
- Substituting the values, a = (2.2^2 - 0^2) / (2 * 3.4) = 1.47 m/s^2.
Step 4: Resolve the force components:
- The horizontal component Fx accelerates the wagon and is related to the acceleration:
Fx = m * a = 12 kg * 1.47 m/s^2 = 17.64 N (to the right or positive direction).
- The vertical component Fy counteracts the gravitational force acting on the wagon:
Fy = m * g, where g is the acceleration due to gravity (9.8 m/s^2).
Fy = 12 kg * 9.8 m/s^2 = 117.6 N (upward).
Step 5: Find the total force applied:
- Since Fx and Fy are at an angle of 38 degrees, we can use trigonometry to find the total force applied F:
F = sqrt(Fx^2 + Fy^2).
Using the Pythagorean theorem, we have:
F = sqrt(17.64^2 + 117.6^2) = 119.64 N.
Therefore, the force applied to the wagon is approximately 119.64 Newtons, at an angle of 38 degrees above the horizontal.
To solve this question, you can use the concepts of Newton's second law and work.
First, let's break down the problem into its components. The force pulling the wagon can be divided into two components: the horizontal component (F_h) and the vertical component (F_v). The angle of 38 degrees above the horizontal can be represented as shown in the figure below.
```
/|
/ |
/ |
/ |
F / | F_v
/ θ |
/ |
/______|__________
F_h
```
Now, let's calculate the net force acting on the wagon. Since the wagon starts from rest and accelerates to a speed of 2.2 m/s, we can use the following equation:
F_net = m * a
where F_net is the net force, m is the mass of the wagon (12 kg), and a is the acceleration. To find the acceleration, we can use the equation:
a = (vf^2 - vi^2) / (2 * d)
where vf is the final velocity (2.2 m/s), vi is the initial velocity (0 m/s), and d is the distance (3.4 m).
Now, let's calculate the force on the wagon. The net force is the vector sum of the horizontal and vertical components of force:
F_net = √(F_h^2 + F_v^2)
To find F_h and F_v, we can use trigonometry.
F_h = F * cos(θ)
F_v = F * sin(θ)
where F is the force we are trying to find and θ is the angle of 38 degrees.
Finally, we can substitute these equations into the net force equation to find F:
F_net = √((F * cos(θ))^2 + (F * sin(θ))^2)
To solve for F, we can rearrange the equation:
F = F_net / √(cos(θ)^2 + sin(θ)^2)
Now, let's plug in the known values into the equation and solve for F:
F = (m * a) / √(cos(θ)^2 + sin(θ)^2)
F = (12 kg * [(2.2 m/s)^2 - (0 m/s)^2] / (2 * 3.4 m)) / √(cos(38°)^2 + sin(38°)^2)
Calculating these values will yield the force applied to the wagon.