A resultant and the direction of acceleration — 9709 P4
Apply the same mass relation to both perpendicular force components.
Syllabus 4.4
Notes and key methods
A constant resultant force gives acceleration in its own direction, with magnitude resultant/mass. For perpendicular component directions, divide each resultant component by the same mass before finding acceleration magnitude.
If a particle starts from rest under a fixed resultant, it then moves along that straight resultant direction. The questions use two perpendicular horizontal directions on a smooth horizontal plane; weight and normal force balance separately. No projectile or curved-trajectory calculation is involved.
Check the conditions
Choose and retain a positive direction; distinguish signed quantities from magnitudes.
Use SI units and \(\displaystyle g= \frac{10m}{s^{2}}\) wherever gravity is relevant.
Apply only the model and time interval stated in the question.
Three worked examples
Example 1 · A resultant and the direction of acceleration
A 10 kg particle starts from rest on a smooth horizontal plane. Constant horizontal forces 8 N east and 5 N north act. Weight and normal reaction balance vertically.
Find the eastward component of acceleration in \(\displaystyle \frac{m}{s^{2}}\).
Find the northward component of acceleration in \(\displaystyle \frac{m}{s^{2}}\).
Find the acceleration magnitude in \(\displaystyle \frac{m}{s^{2}}\), to 3 significant figures.
Find its direction north of east in degrees, to 1 decimal place.
Show the reasoning
(a)
Apply Newton’s second law along the east axis.
Eastward force divided by mass gives \(\displaystyle \frac{4}{5}\) \(\displaystyle \frac{m}{s^{2}}\).
Answer: \(\displaystyle \frac{4}{5}\).
(b)
Apply Newton’s second law along the perpendicular north axis.
Northward force divided by mass gives \(\displaystyle \frac{1}{2}\) \(\displaystyle \frac{m}{s^{2}}\).
Answer: \(\displaystyle \frac{1}{2}\).
(c)
Combine perpendicular acceleration components after dividing by mass.
Pythagoras on the acceleration components gives \(0.943\) \(\displaystyle \frac{m}{s^{2}}\).
Answer: \(0.943\).
(d)
Use the component ratio with the correct reference direction.
The north/east component ratio is \(\displaystyle \frac{5}{8}\), giving 32.0°.
Answer: \(32.0\).
Example 2 · A resultant and the direction of acceleration
A 3 kg particle starts from rest on a smooth horizontal plane. Constant horizontal forces 9 N east and 6 N north act. Weight and normal reaction balance vertically.
Find the eastward component of acceleration in \(\displaystyle \frac{m}{s^{2}}\).
Find the northward component of acceleration in \(\displaystyle \frac{m}{s^{2}}\).
Find the acceleration magnitude in \(\displaystyle \frac{m}{s^{2}}\), to 3 significant figures.
Find its direction north of east in degrees, to 1 decimal place.
Show the reasoning
(a)
Apply Newton’s second law along the east axis.
Eastward force divided by mass gives \(3\) \(\displaystyle \frac{m}{s^{2}}\).
Answer: \(3\).
(b)
Apply Newton’s second law along the perpendicular north axis.
Northward force divided by mass gives \(2\) \(\displaystyle \frac{m}{s^{2}}\).
Answer: \(2\).
(c)
Combine perpendicular acceleration components after dividing by mass.
Pythagoras on the acceleration components gives \(3.61\) \(\displaystyle \frac{m}{s^{2}}\).
Answer: \(3.61\).
(d)
Use the component ratio with the correct reference direction.
The north/east component ratio is \(\displaystyle \frac{2}{3}\), giving 33.7°.
Answer: \(33.7\).
Example 3 · A resultant and the direction of acceleration
A 4 kg particle starts from rest on a smooth horizontal plane. Constant horizontal forces 10 N east and 7 N north act. Weight and normal reaction balance vertically.
Find the eastward component of acceleration in \(\displaystyle \frac{m}{s^{2}}\).
Find the northward component of acceleration in \(\displaystyle \frac{m}{s^{2}}\).
Find the acceleration magnitude in \(\displaystyle \frac{m}{s^{2}}\), to 3 significant figures.
Find its direction north of east in degrees, to 1 decimal place.
Show the reasoning
(a)
Apply Newton’s second law along the east axis.
Eastward force divided by mass gives \(\displaystyle \frac{5}{2}\) \(\displaystyle \frac{m}{s^{2}}\).
Answer: \(\displaystyle \frac{5}{2}\).
(b)
Apply Newton’s second law along the perpendicular north axis.
Northward force divided by mass gives \(\displaystyle \frac{7}{4}\) \(\displaystyle \frac{m}{s^{2}}\).
Answer: \(\displaystyle \frac{7}{4}\).
(c)
Combine perpendicular acceleration components after dividing by mass.
Pythagoras on the acceleration components gives \(3.05\) \(\displaystyle \frac{m}{s^{2}}\).
Answer: \(3.05\).
(d)
Use the component ratio with the correct reference direction.
The north/east component ratio is \(\displaystyle \frac{7}{10}\), giving 35.0°.
Answer: \(35.0\).
Common mistakes
Acceleration follows the resultant force, not necessarily either individual force.