three people are trying to hold a lion still for the vet. The lion, in the center, is wearing a collar with three ropes attached to it and each person has a hold of the rope. charlie is pulling in the direction of 62 degrees west of north with a force of 175 newtons. Sam is pulling in the direction of 43 degrees east of north with a force of 200 newtons. What is the direction and magnitude of the force which must be exerted by Alice on the third rope to counterbalance Sam and Charlie?

Question

three people are trying to hold a lion still for the vet. The lion, in the center, is wearing a collar with three ropes attached to it and each person has a hold of the rope. charlie is pulling in the direction of 62 degrees west of north with a force of 175 newtons. Sam is pulling in the direction of 43 degrees east of north with a force of 200 newtons. What is the direction and magnitude of the force which must be exerted by Alice on the third rope to counterbalance Sam and Charlie?

Answer 1

Charlie: 175 N. @ 152 Deg.,CCW.

Sam: 200 N. @ 47 Deg.,CCW.

X = 175*cos152 + 200*cos47 = -18.1 N.
Y = 175*sin152 + 200*sin47 = 228.4 N.

Alice: Force must be equal and opposite
The vector sum of Charlie and Sam's force.

X = 18.1 N.
Y = -228.4 N.

tanA = Y/X = -228.4 / 18.1 = -12.61878.
A = -85.5 Deg. = 85.5 Deg S of E
A = -85.5 + 360 = 274.5 Deg.,CCW..

M = X / cosA = 18.1 / cos274,5 = 230.7 N.@ 274.5 Deg.

Answer 2

To find the direction and magnitude of the force that Alice must exert, we can break it down into its horizontal and vertical components.

First, let's calculate the horizontal and vertical components of Charlie's force:
Horizontal component: 175 N * cos(62°)
Vertical component: 175 N * sin(62°)

Next, let's calculate the horizontal and vertical components of Sam's force:
Horizontal component: 200 N * cos(43°)
Vertical component: 200 N * sin(43°)

Now, let's sum up the horizontal and vertical components of Charlie and Sam's forces:
Horizontal component: Charlie's horizontal component + Sam's horizontal component
Vertical component: Charlie's vertical component + Sam's vertical component

Finally, we can use these horizontal and vertical components to find the magnitude and direction of the force that Alice must exert:
Magnitude of the force: sqrt(horizontal component^2 + vertical component^2)
Direction of the force: arctan(vertical component / horizontal component)

Let's calculate the exact values:

Horizontal component (Charlie): 175 N * cos(62°) = -92.31 N
Vertical component (Charlie): 175 N * sin(62°) = 151.64 N

Horizontal component (Sam): 200 N * cos(43°) = 144.67 N
Vertical component (Sam): 200 N * sin(43°) = 136.64 N

Horizontal component (Charlie + Sam): -92.31 N + 144.67 N = 52.36 N
Vertical component (Charlie + Sam): 151.64 N + 136.64 N = 288.28 N

Magnitude of the force (Alice): sqrt(52.36^2 + 288.28^2) = 294.85 N
Direction of the force (Alice): arctan(288.28 N / 52.36 N) = 80.2°

Therefore, Alice must exert a force of approximately 294.85 N at an angle of 80.2° east of north to counterbalance the forces exerted by Sam and Charlie.