Question

how fast will and in what direction will a 20kg object accelerate if one force pushes at a 30 degree angle and another pushes at a 120 degree angle? these angles are principal angles.​

Answer 1

`|a|=2.83\ m/s^2`

`\theta=75^o`

Step-by-step explanation:

Net Force And Acceleration

The Newton's second law relates the net force applied on an object of mass m and the acceleration it aquires by

`\vec F_n=m\vec a`

The net force is the vector sum of all forces. In this problem, we are not given the magnitude of each force, only their angles. For the sake of solving the problem and giving a good guide on how to proceed with similar problems, we'll assume both forces have equal magnitudes of F=40 N

The components of the first force are

`\vec F_1=<40cos30^o,40sin30^o>`

`\vec F_1=<20√(3),20>\ N`

The components of the second force are

`\vec F_2=<40cos120^o,40sin120^o>`

`\vec F_2=<-20,20√(3)>\ N`

The net force is

`\vec F_n=<20√(3)-20,20+20√(3)>`

`\vec F_n=<14.64,54.64>\ N`

The magnitude of the net force is

`|F_n|=√(14.64^2+54.64^2)`

`|F_n|=√(3200)=56.57\ N`

The acceleration has a magnitude of

`\displaystyle |a|=(|F_n|)/(m)`

`\displaystyle |a|=(56.57)/(20)`

`|a|=2.83\ m/s^2`

The direction of the acceleration is the same as the net force:

`\displaystyle tan\theta=(54.64)/(14.64)`

`\theta=75^o`