Two forces, one with a magnitude of 16 N and the other with a magnitude of 29 N are the only forces acting on a 8.5 kg object.
What is the maximum possible net acceleration that could act on the 8.5 kg object? Answer in units of m/s^2.
What is the minimum possible net acceleration that could act on the 8.5 kg object? Answer in units of m/s^2.
If the acceleration of the object has a magnitude of 3.6 m/s^2, what is the angle between the two forces? Answer in units of degree.
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When two forces act on an object simultaneously:
The magnitude of resultant force is maximum when the two force act in the same direction. The magnitude of this resultant force is equal to the sum of magnitude these two forces.
And the magnitude of resultant force is minimum when the two force act in the opposite direction. The magnitude of this resultant force is equal to the difference in magnitude of these two forces.
Magnitude of two forces acting on the object are:
f1 = 16 N and f2 = 29 N
Mass of the object = m =8.5 kg
Maximum resultant force acting on object = F(max) = f1 + f2 = 16 + 29 = 45 N
Minimum resultant force acting on object = F(min) = f2 - f1 = 16 - 29 = - 13 N
Maximum acceleration = F(max)/m = 45/8.5 = 5.2941 m/s^2
Maximum accleration = F(max)/m = -13/8.5 = -1.5294 m/s^2
Calculating angle (A) between forces when the acceleration is 3.6 m/s^2
when the acceleration = a = 3.6 m/s^2
resultant force = f = a*m = 3.6*8.5 = 30.6
But: f = (f1^2 + f2^2 + 2f1*f2*Cos A)^1/2
Therefore: 30.6 = (16^2 + 29^2 + 2*16*29*Cos A)^1/2
Therefore: (30.6)^2 = (256 + 841 + 928*Cos A)
Therefore: 936 = (1097 + 928*Cos A)
Therefore Cos A = (936 - 1097)/928 = -0.17349
Therefore A = 170 degrees
Magnitude of maximum possible acceleration = 5.2941 m/s^2
Magnitude of minimum possible acceleration = 1.5294 m/s^2 (in opposite direction)
Angle between forces when acceleration is 3.3 m/s^2 = 170 degrees.
The law parallelogram of forces states the forces a and b with angle x between them acting has a resultant force c given by:
c = sqrt(a^2+b^2+2bc*cos x).
Therefore, c is maximum if x= 0 , when c= a+b. c miminmum if x=180, when c=a-b.
Therefore the maximum and minimum accelerations are:
(29+16)N/ 8.5=5.2941 m/s^2 when the forces are in the same direction or angle x=0 and
(29-16)/8.6 = 1.52942 m.s^2 when forces are in opposite direction or x=180 deg and this acceleration is in the direction of the greater force i.e, 29N.
If the acceleration is 3.6m/s^2, the n the force = 8.5*3.6= 30.6N . Or c^2=29.5^2 =29^2+16^2+2*29*16cosx or
cosx = (30.6^2-29^2-16^2)/(2*29*16) = -0.173103448
Therefore x= cos inverse(-0.173103448)= 99.9683 degree.
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