1. Three astronauts, propelled by jet backpacks, push and guide a 119 kg asteroid toward a processing dock, exerting the forces shown in the figure, with F1 = 30 N, F2 = 57 N,F3 = 36 N, θ1 = 30°, and θ3 = 60°. What is the (a) magnitude and (b) angle (measured relative to the positive direction of the x axis in the range of (-180°, 180°]) of the asteroid’s acceleration?

2. Tarzan, who weighs 808 N, swings from a cliff at the end of a 18.4 m vine that hangs from a high tree limb and initially makes an angle of 19.7° with the vertical. Assume that an xaxis points horizontally away from the cliff edge and a y axis extends upward. Immediately after Tarzan steps off the cliff, the tension in the vine is 761 N. Just then, what are (a) the force from the vine on Tarzan in unit-vector notation, and (b) the net force acting on Tarzan in unit-vector notation? What are (c) the magnitude and (d) the direction (measured counterclockwise from the positive x-axis) of the net force acting on Tarzan? What are (e) the magnitude and (f) the direction of Tarzan’s acceleration just then?

3. An elevator and its load have a combined mass of 1800 kg. Find the tension in the supporting cable when the elevator, originally moving downward at 13 m/s, is brought to rest with constant acceleration in a distance of 26 m.

23490 Units: N

4. Holding onto a tow rope moving parallel to a frictionless ski slope, a 78.3 kg skier is pulled up the slope, which is at an angle of 8.0° with the horizontal. What is the magnitude Frope of the force on the skier from the rope when (a) the magnitude v of the skier’s velocity is constant at 1.77 m/s and (b) v = 1.77 m/s as v increases at a rate of 0.109 m/s2?

a. 106.79 Units: N

b. 115.33 Units: N

5. In the figure, a horse pulls a barge along a canal by means of a rope. The force on the barge from the rope has a magnitude of 7870 N and is at the angle θ = 17° from the barge’s motion, which is in the positive direction of an x axis extending along the canal. The mass of the barge is 9500 kg, and the magnitude of its acceleration is 0.12 m/s2. What are (a) the magnitude and (b) the direction (measured from the positive direction of the x axis) of the force on the barge from the water? Give your answer for (b) in the range of (-180°, 180°].

6. The figure here shows four penguins that are being playfully pulled along very slippery (frictionless) ice by a curator. The masses of three penguins and the tension in two of the cords are m1 = 12 kg, m3 = 15 kg, m4 = 23 kg, T2 = 109 N, and T4 = 218 N. Find the penguin mass m2 that is not given.

7. The figure shows a man sitting in a bosun’s chair that dangles from a massless rope, which runs over a massless, frictionless pulley and back down to the man’s hand. The combined mass of man and chair is 63.9 kg. With what force magnitude must the man pull on the rope if he is to rise (a) with a constant velocity and (b) with an upward acceleration of 1.10 m/s2? (Hint: A free-body diagram can really help.) Problem continues below.

If the rope on the right extends to the ground and is pulled by a co-worker, with what force magnitude must the co-worker pull for the man to rise (c) with a constant velocity and (d)with an upward acceleration of 1.10 m/s2? What is the magnitude of the force on the ceiling from the pulley system in (e) part a (f) part b, (g) part c, and (h) part d?

8. The figure shows a box of mass m2 = 1.3 kg on a frictionless plane inclined at angle θ = 29°. It is connected by a cord of negligible mass to a box of mass m1 = 3.4 kg on a horizontal frictionless surface. The pulley is frictionless and massless. (a) If the magnitude of the horizontal force is 1.7 N, what is the tension in the connecting cord? (b) What is the largest value the magnitude of may have without the connecting cord becoming slack?

9. The figure shows a section of a cable-car system. The maximum permissible mass of each car with occupants is 2300 kg. The cars, riding on a support cable, are pulled by a second cable attached to the support tower on each car. Assume that the cables are taut and inclined at angle θ = 36°. What is the difference in tension between adjacent sections of pull cable if the cars are at the maximum permissible mass and are being accelerated up the incline at 0.77 m/s2?

10. A worker drags a crate across a factory floor by pulling on a rope tied to the crate. The worker exerts a force of magnitude F = 470 N on the rope, which is inclined at angle θ = 31° to the horizontal, and the floor exerts a horizontal force of magnitude f = 130 N that opposes the motion. Calculate the magnitude of the acceleration of the crate if (a) its mass is 300 kg and (b) its weight is 300 N.

a. 0.91 Units: m/s^2 b. 8.91 m/s^2