A fire tug discharges salt water (density 1030 kg/m^3) with a nozzle velocity of 40 m/s at a rate of 0.080 m^3/s. The discharge is angled 30 degrees above the horizontal. To maintain a fixed position, what propeller thrust T must the tug develop?
Explanation
This problem is an application of the steady-flow momentum equation (Section 4/6) to determine the reaction force from a water jet. The horizontal thrust must balance the horizontal component of the momentum flux of the discharged water.
Other questions
According to the generalized Newton's second law for a system of particles, the resultant of the external forces on the system equals the total mass of the system times what quantity?
For a system of particles, the sum of all internal forces is identically zero. Why is this the case?
The total kinetic energy of a mass system can be expressed as the sum of two components. What are these two components?
What is the linear momentum, G, of a system of particles with constant mass 'm' and mass-center velocity 'v_bar'?
In the analysis of steady mass flow through a rigid container, what does the term 'm_dot * Δv' in the equation ΣF = m_dot * Δv represent?
For a system with variable mass, such as a rocket, what does the term 'm_dot * u' in the equation of motion ΣF = m * v_dot + m_dot * u represent?
Under what condition is the total linear momentum of a system of particles conserved?
Under what condition is the total angular momentum of a system of particles about its mass center, G, conserved?
A system of three particles with masses m, 2m, and 4m has a single external force F applied to the 4m particle in the positive y-direction. The system rests on a smooth horizontal surface. What is the acceleration of the system's mass center?
In the analysis of steady mass flow, the force-momentum equation is ΣF = m_dot(v_out - v_in). What does the term ΣF represent in this context?
A jet aircraft with a mass of 4.6 Mg has a drag of 32 kN at a speed of 1000 km/h. It consumes air at 106 kg/s and fuel at 4 kg/s, with an exhaust velocity of 680 m/s relative to the nozzle. To maintain a constant speed of 1000 km/h, what is the maximum angle of elevation at which the jet can fly?
A jet ski has reached its maximum velocity of 70 km/h. Its pump discharges salt water (density 1025 kg/m^3) from a horizontal 50-mm diameter nozzle at a rate of 0.082 m^3/s. The water enters the intake at the ski's velocity. What is the resistance R of the water to the hull at this speed?
An empty 50-kg tank is propelled by a force P while scooping up fresh water from a stream flowing at 1.5 m/s in the opposite direction. At an instant when the tank contains 80 kg of water and is moving at 2 m/s with an acceleration of 0.4 m/s^2, what is the required force P? The scoop entrance area is 2000 mm^2.
A rocket with an initial total mass m0 is fired vertically. It burns fuel at a constant rate, and the exhaust gas has a constant relative velocity u. If the residual mass is mb, what is the maximum velocity reached by the rocket, neglecting air resistance?
The end of a chain of length L and mass per unit length rho is lifted vertically from a piled platform with a constant velocity v. What is the force P required as a function of the height x of the end above the platform?
A 450-kg ram of a pile driver falls 1.4 m from rest and strikes a 240-kg pile. The ram and pile move together with no rebound. What is the velocity of the pile and ram immediately after impact?
An 8-Mg truck is at rest on the deck of a 240-Mg barge, which is also at rest in still water. If the truck starts and drives toward the bow at a speed of 6 km/h relative to the barge, what is the resulting speed of the barge?
A 100-kg boulder moving at 1.2 m/s at a 30-degree angle hits a stationary 300-kg mine car. After the impact, they become coupled. The car was rolling at 0.6 m/s towards the impact. What is the final velocity of the coupled system?
What is the primary reason for analyzing the angular momentum of a system about the mass center G (HG) rather than just a fixed point O (HO)?
For the generalized Newton's second law ΣF = ma to be valid, what is a key requirement for the reference axes from which the acceleration 'a' of the mass center is measured?
When is it permissible to use the simplified force-momentum equation ΣF = d(mv)/dt for a system whose mass is changing?
A 10-kg block is at rest on a rough incline (angle 20 degrees, static friction 0.50, kinetic friction 0.40). At t=0, a time-varying force P, directed up the incline, is applied. P increases linearly from 0 to 100 N over 6 seconds. What is the velocity of the block at t = 3 seconds?
Two freight cars, A (130,000 lb) and C (150,000 lb), are rolling at 2 mi/hr and 1.5 mi/hr respectively. They are uncoupled. What is the magnitude of the total linear momentum of this two-car system?
Which principle is most directly applicable for analyzing the velocity change of a system over a specified time interval when forces are known as functions of time?
A 12-Mg truck drives onto a 350-Mg barge at 20 km/h and brakes to a stop on the deck. The barge is initially at rest in calm water. What is the final speed of the barge?
Why is the term 'm_dot * v' often referred to as 'momentum flux' in the context of steady mass flow?
In the variable-mass equation of motion, ΣF = m*dv/dt + m_dot*u, what does the variable 'u' represent?
Three identical spheres are welded to a light rigid frame rotating in a horizontal plane about a vertical axis through O. A couple M_O = 30 N-m is applied for 5 seconds to the frame, which is initially rotating at 20 rad/s. What is the new angular velocity? Each sphere has a mass of 4 kg and is at a radius of 0.5 m, another at 0.4 m, and the last at 0.6 m.
The angular momentum of a system of particles about an arbitrary, possibly accelerating, point P is given by HP = HG + rho_bar x (m*v_bar), where HG is the angular momentum about the mass center G. What does the term rho_bar x (m*v_bar) represent?
A two-stage rocket is in space when the first stage burns out and the 1400 kg second stage ignites. The second stage burns fuel at 5.2 kg/s with an exhaust velocity of 3000 m/s relative to its nozzle. What is the acceleration of the second stage 60 seconds after ignition, assuming a constant g of 8.70 m/s^2?
A rotary snow plow moves at a constant 20 km/h and discharges 60 Mg of snow per minute with a velocity of 12 m/s relative to the plow, at an angle of 45 degrees. What is the tractive force P on the tires required to maintain the constant speed?
If a system is conservative and no external forces (other than potential forces like gravity) do work on it, which quantity is conserved?
The total linear momentum of a system of five particles at time t = 2.2 s is G = 3.4i - 2.6j + 4.6k kg-m/s. At t = 2.4 s, the momentum is G = 3.7i - 2.2j + 4.9k kg-m/s. What is the magnitude of the time average of the resultant external force on the system during this interval?
A 4-kg cart is at rest. It is acted on by a horizontal force that is initially 20 N and increases linearly to 30 N over 4 seconds. Neglecting friction, what is the velocity of the cart at t = 3 seconds?
A 1000-kg car is moving at 30 km/h. It accelerates uniformly to 70 km/h over 15 seconds. The average rolling resistance is 500 N. What is the average tension in the tow cable pulling the car?
A 1.4 Mg test vehicle is accelerated from rest by a high-velocity water jet (v = 150 m/s) from a 140-mm-diameter nozzle. Frictional resistance is 10 percent of the vehicle's weight. What is the velocity of the vehicle 3 seconds after release?
An ideal rope of length L and mass per unit length rho is pulled by a force P, giving end A a constant velocity v. The rope is doubled back on itself, with end B fixed. What is the force P required?
The principle ΣF = G_dot (where G is linear momentum) is an alternative form of which fundamental law?
A 0.3 kg block is coated with a weak adhesive and pressed onto a turntable. The turntable starts from rest at t=0 and accelerates uniformly with alpha = 2 rad/s^2. If the adhesive fails at t = 3 s, what was the ultimate shear force supported by the adhesive? The block is 0.5 m from the center.
A system consists of two 2-kg balls on a smooth horizontal surface. They are initially at rest. A vertical force of 60 N is applied to the junction of the wires connecting them. What is the initial vertical acceleration of each ball?
The work-energy equation for a system of particles, U_1-2 = ΔT, relates the total work done to the change in total kinetic energy. What forces are included in the work term U_1-2?
A 1500-kg car has a velocity of 30 km/h up a 10-percent grade. The driver applies power for 8 seconds to reach 60 km/h. What is the time average of the total tangential force F exerted on the tires during this period?
An empty 25 Mg coal car is moving at 1.2 m/s when a hopper begins releasing coal into it at a rate of 4 Mg per second. How far has the car moved after 32 Mg of coal have been deposited?
A 1200-kg car, initially at rest, is towed by a cable. The car accelerates uniformly from 30 km/h to 70 km/h over a 15-s interval. What is the average power transmitted by the tow cable during this interval, assuming the cable is horizontal and rolling resistance is a constant 500 N?
An open-link chain of length L and mass rho per unit length is released from rest with a small overhang x to initiate motion over a smooth corner. What is the velocity of the chain when the last link A reaches the corner?
A jet of water is deflected by a fixed 60-degree vane. If fresh water (1000 kg/m^3) issues from a nozzle at 30 m/s at a rate of 0.05 m^3/s and is split into two equal streams, what is the force F required to hold the vane?
A 100-lb block is initially stationary on a surface where the static friction coefficient is 0.60 and the kinetic friction is 0.40. A force P, which increases linearly from 0 to 80 lb over 4 seconds, is applied horizontally. What is the speed of the block at t = 4 seconds?
An 88-kg sprinter reaches a maximum speed of 11 m/s in 2.5 s with uniform acceleration, starting from rest. What is his power output at the instant his speed is 5 m/s?
A 900-kg motorized unit A is used to slow the descent of a 600-kg bucket B from 3 m/s to 0.5 m/s over 6 seconds. What is the average force R which the cable system exerts to support unit A during this time?