Forces Not at 90°
Finding the Resultant & Equilibrant When Concurrent Forces Meet at Any Angle
Angle ≠ 90° — when concurrent forces don't meet at a right angle, the resultant can't be found with the Pythagorean theorem. It must be found graphically.
F. Resultants When the Angle Between Forces ≠ 90°
5 N North & 3 N 30° S of E act concurrently (on a single point).
Find the resultant & equilibrant.
Angle not 90 degrees, can't use Pythagorean theorem
Graphical Method
1. Draw vectors to scale
2. Make parallelogram
3. Draw diagonal (resultant)
4. Find magnitude & direction of vector
Graphical Method
1. Measure length, convert to Newtons using scale 1 cm = 1 N
2. Measure angle vector makes with horizontal
Try This at Home
You can do this with a ping pong ball and a hair dryer
St. Mary's U. Astronomy and Physics Dept.
When a fluid moves in a single direction, it exerts very little pressure perpendicular to the direction of its flow. The external air pressure keeps the ball in place.
Check Your Understanding
5 quick questions — pick an answer to see if you’re right.
1. Why can't the Pythagorean theorem find the resultant of two forces at 30° to each other?
2. In the parallelogram method, after drawing both vectors to scale from one point, the resultant is
3. Using a scale of 1 cm = 1 N, a resultant drawn with the graphical method measures 4.5 cm. What is its magnitude?
4. A resultant is 4.5 N, 53° N of E. What is the equilibrant?
5. Forces of 5 N and 3 N act on a point at an angle other than 0°, 90° or 180°. Which resultant is possible?
Two Ropes on a Sled
Two students pull a sled across flat snow using two ropes. Each student pulls with a horizontal force of 50. N. The angle between the two ropes is 60.°, and the sled moves along the line that splits this angle in half.
Which value is closest to the magnitude of the resultant of the two forces?
▶ Want a challenge? Open 6 more Regents-style questions (harder · mixed formats)
Challenge score: 0 / 6
C1. Each 50.-N rope force can be split into a component along the sled’s path and a component perpendicular (sideways) to it. Determine the magnitude of the sideways component of one rope’s force.
C2. If the angle between the ropes is increased to 120°, what is the resultant?
C3. A student makes the following claim.
Place a check mark in either the Support or Refute box. Justify your response using a mathematical relationship.
C4. The students move closer together so the angle between the ropes is 40.°. Each still pulls with 50. N. Substitute numerical values into a mathematical representation that can be used to determine the magnitude of the resultant. Then calculate the resultant.
C5. Which list ranks the resultant from greatest to least for angles of 0°, 60.°, 90.°, and 120°?
C6. The students want the largest possible pull on the sled, but if their ropes are too close together they bump into each other. Identify one benefit and one drawback of increasing the angle between the ropes.
