john and paul are trying to move a 100kg piano which is resulting on a flat, smooth floor. John pushing towards the North, while Paul is pushing towards the East. Both are exerting a 100N force. What is acceleration of the piano (magnitude and direction)?

A=200/100=2m/s^2

45 North East

110^2+100^2=20,000

Sqrt*(20,000)=141N

141N=m*a

141N=100kg*a

141N/100kg=a

a=1.41m/s^2

Direction:

45 North East

110^2+100^2=20,000

Magnitude:

Sqrt*(20,000)=141N

Acceleration:

141N=m*a

141N=100kg*a

141N/100kg=a

a=1.41m/s^2

45 North East

110^2+100^2=20,000

Sqrt*(20,000)=141N

Not sure how you came up with 45NE and 110? Did you add force of gravity?

Trish,

Its vector addition:

--->
^
I
I
I

Spacing is a problem on this form. But one vector is pointing up (north) and the other is pointing to the right (east). Intuitively, you should be able to see which way the resultant vector will point. However, you may want to go over or google head to tail method, to see which way the resultant vector will point. Since magnitudes are equal, you should already know that the resultant angle will be between the north and east vector.

I have a typo: it should read 100^2+100^2=20,000 and not 110^2+100^2=20,000

The math is based on the Pythagorean theorem.

x^2+y^2=r^2 or 110^2+100^2=20,000

Solving for r:

r=sqrt*(x^2+y^2)

To find the acceleration of the piano, we can use Newton's second law of motion, which states that the acceleration of an object is equal to the net force acting on it divided by its mass. In this case, the net force is the combination of the forces exerted by John and Paul.

First, let's analyze the forces in each direction. John is pushing towards the North, creating a force in the positive Y-direction, and Paul is pushing towards the East, creating a force in the positive X-direction. Since these forces are perpendicular to each other, we can treat them independently.

The force exerted by John is 100N in the positive Y-direction, and the force exerted by Paul is 100N in the positive X-direction. The total force in the X-direction is 100N, and in the Y-direction is also 100N.

Now, let's find the acceleration in each direction separately.

In the X-direction:
The force in the X-direction is 100N.
The mass of the piano is given as 100kg.
Using Newton's second law (F=ma), we can rearrange the formula to calculate the acceleration (a=F/m):
a_x = F_x/m = 100N/100kg = 1m/s^2

In the Y-direction:
The force in the Y-direction is 100N.
The mass of the piano is given as 100kg.
Using Newton's second law (F=ma), we can rearrange the formula to calculate the acceleration (a=F/m):
a_y = F_y/m = 100N/100kg = 1m/s^2

Now, to find the total acceleration, we can use the Pythagorean theorem, which states that the magnitude of the total acceleration is equal to the square root of the sum of the squares of the components:
a = √(a_x^2 + a_y^2)
a = √(1^2 + 1^2)
a = √2 m/s^2 (magnitude)

The direction of the acceleration can be determined by finding the angle between the resultant acceleration and the positive X-axis. This can be found using trigonometry:
θ = arctan(a_y / a_x)
θ = arctan(1/1)
θ = 45° (direction)

Therefore, the magnitude of the acceleration is √2 m/s^2, and the direction is 45° with respect to the positive X-axis.