Answer: To determine the pulling force P, we can use the conservation of energy principle, which states that the change in the kinetic energy of an object is equal to the work done on it by external forces.
At position 1, the slider has a velocity of vi = 0.4 m/s and is at a height of h1 = 200 mm = 0.2 m above the ground. At position 2, the slider has a velocity of v2 = 0.6 m/s and is at a height of h2 = 250 mm = 0.25 m above the ground. The angle of the incline is θ = 15°.
The change in kinetic energy of the slider is:
ΔK = (1/2) m (v2^2 - vi^2)
where m is the mass of the slider, which is given as 10 kg.
ΔK = (1/2) (10 kg) ((0.6 m/s)^2 - (0.4 m/s)^2)
ΔK = 1.2 J
The work done on the slider by external forces is equal to the sum of the work done by the pulling force P and the work done by the spring force Fs. The work done by the pulling force is:
Wp = P (s2 - s1)
where s1 and s2 are the elongations of the spring at positions 1 and 2, respectively. The elongation of the spring is given by:
s = l - l0
where l is the length of the spring and l0 is its unstretched length, which is given as 200 mm = 0.2 m.
At position 1, the elongation of the spring is:
s1 = l1 - l0 = h1/sinθ - l0
s1 = 0.2 m / sin 15° - 0.2 m
s1 = 0.5828 m
At position 2, the elongation of the spring is:
s2 = l2 - l0 = h2/sinθ - l0
s2 = 0.25 m / sin 15° - 0.2 m
s2 = 0.8328 m
Therefore, the work done by the pulling force is:
Wp = P (s2 - s1)
Wp = P (0.25 m/sin15° - 0.2 m/sin15°)
Wp = 0.1077 P J
The work done by the spring force is:
Ws = (1/2) k (s2^2 - s1^2)
where k is the stiffness of the spring, which is given as 400 N/m.
Ws = (1/2) (400 N/m) ((0.8328 m)^2 - (0.5828 m)^2)
Ws = 44.56 J
The total work done on the slider is:
W = Wp + Ws
W = 0.1077 P + 44.56 J
According to the conservation of energy principle, this must be equal to the change in kinetic energy of the slider:
W = ΔK
0.1077 P + 44.56 J = 1.2 J
Solving for P, we get:
P = (1.2 J - 44.56 J) / 0.1077
P = -370.9 N
The negative sign indicates that the pulling force is in the opposite direction to the motion of the slider. Therefore, the magnitude of the pulling force required to produce the observed velocities is 370.9 N.
Explanation: