This is for physics class.
Sally applies 32 N of force at an angle of 22 degrees above the horizontal to a 12 kg crate.
Ignoring friction, calculate the horizontal acceleration of the crate?
Conceptual: will the acceleration of the crate increase or decrease if the same force is applied at an angle of 16 degrees? Explain.
Start with a picture if you don't already have one and then draw a free body diagram (a dot with arrows to represent all the forces applied to the box if you don't know what that is go ahead and google it- it'll make sense if you see a picture). We can assume up and the direction of the applied force (F) is positive.
Forces in the y direction:
+Fn (normal force)
-mg (mass*gravity)
-F*sin(theta)
Fsin(theta) is the componet of the applied force in the y direction. If you remember the trig identites sin(theta)=opposite/hypotenuse. Here, the applied force is the hypo. so you get Fsin(theta)= the y component of the applied force since it is ppposite the angle at which the force is applied to the box.
X direction:
+Fcos(theta)
This is the same as the y component of the applied force except its the adjacent side to the angle and acts in the x direction.
Ok if that all made sense (I hope it did) now you're ready to write two equations, one in the x direction and one in the y, based on Newton's second law F=ma
Y direction:
Fn-mg-Fsin(theta)=ma
X direction:
Fcos(theta)=ma
Since you have all the information in the x direction substitute with numbers and solve for a. It should come out to be something around 2.47 m/s^2 if you rounded the same way I did.
PS if there was friction you would subtract its force from the x direction when using sum of forces = ma.
Hope this helps!
To determine the horizontal acceleration of the crate, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F = m*a).
In this case, Sally applies a force of 32 N at an angle of 22 degrees above the horizontal to a 12 kg crate. We need to find the horizontal component of this force, as it is the force responsible for the crate's acceleration in the horizontal direction.
To find the horizontal component of the force, we can use the equation:
F_horizontal = F * cos(theta)
where F is the applied force and theta is the angle between the applied force and the horizontal direction.
So, plugging in the values:
F_horizontal = 32 N * cos(22 degrees)
F_horizontal = 32 N * 0.92718385
F_horizontal = 29.545 N
Now that we have the horizontal component of the force, we can calculate the acceleration using Newton's second law:
F_horizontal = m * a
Substituting the values:
29.545 N = 12 kg * a
Solving for acceleration (a):
a = 29.545 N / 12 kg
a = 2.462 m/s^2
So, the horizontal acceleration of the crate is 2.462 m/s^2.
Now let's address the conceptual question regarding the effect of changing the angle on the acceleration.
If the same force (32 N) is applied at an angle of 16 degrees instead of 22 degrees, the horizontal component of the force would decrease.
To calculate the new horizontal component:
F_horizontal_new = 32 N * cos(16 degrees)
F_horizontal_new = 32 N * 0.9612617
F_horizontal_new = 30.762 N
As the horizontal component of the force increases, the acceleration would also increase. This is because the force responsible for the horizontal acceleration is larger when the angle is smaller.
Therefore, if the same force is applied at an angle of 16 degrees instead of 22 degrees, the acceleration of the crate would increase.