.A climber of mass 80 kg is on a steep rock face. The force X
that the rock exerts on the climber is at an angle of 50° to
the vertical. Y, the other force on the climber, keeps him in
equilibrium and is provided by a rope at an angle of 40° to the
vertical.
a) Draw a sketch to show the forces acting on the climber.
b) From your sketch of the forces, sketch a triangle of forces to
show equilibrium.
c) Use your triangle of forces to find
i) X, the force the rock face exerts on the climber
ii) Y, the force provided by the rope.
a) Sketch:
^
|
Y |
| |
| |
| | X
| |
| |
-----------
b) Triangle of forces for equilibrium:
^
/|
/ |
/ |
/ Y |
/ |
/ |
------X
c) i) To find X, we can use the triangle of forces. The vertical component of the force X can be found by using the sine of the angle (50°) and multiplying it by the magnitude of X:
Vertical component of X = X * sin(50°)
Since the climber is in equilibrium, the vertical component of X should be equal to the vertical component of Y, which is given by:
Vertical component of Y = Y * sin(40°)
Setting these two equal to each other, we have:
X * sin(50°) = Y * sin(40°)
ii) To find Y, we can use the triangle of forces. The horizontal component of Y can be found by using the cosine of the angle (40°) and multiplying it by the magnitude of Y:
Horizontal component of Y = Y * cos(40°)
Since the climber is in equilibrium, the horizontal component of Y should be equal to the horizontal component of X, which is given by:
Horizontal component of X = X * cos(50°)
Setting these two equal to each other, we have:
Y * cos(40°) = X * cos(50°)
a) Sketch:
```
X
| /
|/
|
|---Climber---
|
|
Y|
|
```
Here, X represents the force exerted by the rock face on the climber, and Y represents the force provided by the rope.
b) Equilibrium Triangle Sketch:
```
X
| /
|/
|
|---Climber---
| \
| \
Y|___\
```
In this triangle, X and Y are the two forces acting on the climber, and the climber is in equilibrium.
c) i) To find X, we can use trigonometry:
We have a right triangle with X as the hypotenuse. The angle between X and the vertical is given as 50°. Now, we can use trigonometric relations to find X.
Using the cosine function:
cos(50°) = Adjacent / Hypotenuse
cos(50°) = Y / X
Solving for X:
X = Y / cos(50°)
ii) To find Y, we can use trigonometry as well:
We have a right triangle with Y as the hypotenuse. The angle between Y and the vertical is given as 40°. Similar to the previous step, we can use trigonometric relations to find Y.
Using the cosine function:
cos(40°) = Adjacent / Hypotenuse
cos(40°) = X / Y
Solving for Y:
Y = X / cos(40°)
Therefore, we can determine the forces as follows:
i) X = Y / cos(50°)
ii) Y = X / cos(40°)
Note: Once you have the values of X and Y from the equations above, substitute the given values for X and Y to calculate the actual forces exerted by the rock face and the rope on the climber.