Two tugboats are towing a ship.Each exerts a force of 6000 N and the angle between two ropes is 60 degrees.What is the resultant force on the ship?
10392 N
10400
Two tugboats are towing a ship.Each exerts a force of 6000 N and the angle between two ropes is 60 degrees.What is the resultant force on the ship?
Physics
Answer
To determine the resultant force on the ship, we need to find the vector sum of the forces exerted by the two tugboats. The force from each tugboat can be represented as a vector, with the magnitude of 6000 N and the angle between the ropes being 60 degrees.
To find the resultant force, we can use vector addition:
1. Start by representing the first force as a vector. Since the angle between the ropes is given as 60 degrees, we can draw a vector pointing in the direction of the force and label it as 6000 N.
2. Next, represent the second force as a vector. Since the angle between the ropes is also 60 degrees, draw another vector pointing in the direction of the force and label it as 6000 N.
3. Place the vectors tip-to-tail, connecting the tail of the first vector to the tip of the second vector. The resulting vector represents the resultant force.
4. Measure the magnitude and direction of the resultant vector. The magnitude can be found using trigonometry. Since the two forces are equal, the angle between them is 60 degrees, and the triangle formed is an equilateral triangle. Therefore, the magnitude of the resultant force is equal to twice the magnitude of either of the individual forces.
Resultant force magnitude = 2 * 6000 N = 12000 N
5. Since the forces are acting at the same angle, the direction of the resultant force is the same as the direction of the individual forces. Therefore, the resultant force acts in the same direction as the tugboats.
Resultant force direction = 60 degrees
So, the resultant force on the ship is 12000 N directed at 60 degrees.
F(net)= 2F•cos30°
Fr = 6000[0o] + 6000[60o]
Fr = 6000 + 3000+5196i = 9000 + 5196i =
Fr = Sqrt(9000^2 + 5196^2).
We need to use law of parallelogram of vector addition. For two vectors
→
P
and
→
Q
having an angle
θ
between the two the resultant vector
→
R
is given by
∣
∣
∣
→
R
∣
∣
∣
=
√
∣
∣
∣
→
P
∣
∣
∣
2
+
∣
∣
∣
→
Q
∣
∣
∣
2
+
2
∣
∣
∣
→
P
∣
∣
∣
∣
∣
∣
→
Q
∣
∣
∣
cos
θ
and angle of the resultant
α
=
tan
−
1
⎛
⎜
⎜
⎜
⎝
∣
∣
∣
→
Q
∣
∣
∣
sin
θ
∣
∣
∣
→
P
∣
∣
∣
+
∣
∣
∣
→
Q
∣
∣
∣
cos
θ
⎞
⎟
⎟
⎟
⎠
Inserting given values we get
∣
∣
∣
→
R
∣
∣
∣
=
√
(
6000
)
2
+
(
6000
)
2
+
2
×
6000
×
6000
×
cos
60
⇒
∣
∣
∣
→
R
∣
∣
∣
=
√
(
6000
)
2
+
(
6000
)
2
+
2
×
6000
×
6000
×
0.5
⇒
∣
∣
∣
→
R
∣
∣
∣
=
6000
√
3
⇒
∣
∣
∣
→
R
∣
∣
∣
=
10392.3
N
rounded to one decimal place.
and
α
=
tan
−
1
(
6000
×
sin
60
6000
+
6000
cos
60
)
⇒
α
=
tan
−
1
⎛
⎜
⎝
√
3
2
1
+
1
2
⎞
⎟
⎠
⇒
α
=
tan
−
1
(
1
√
3
)
⇒
α
=
30
∘
measured as per measurement of
∠
θ