Classical Mechanics Answers

In the figure below, a spring with k = 170 N/m is at the top of a frictionless incline of angle θ = 370. The lower end of the incline is distance D = 1.00 m from the end of the spring, which is at its relaxed length. A 2.00 kg canister is pushed against the spring until the spring is compressed 0.200 m and released from rest. (a) What is the speed of the canister at the instant the spring returns to its relaxed length (which is when the canister loses contact with the spring)? (b) What is the speed of the canister when it reaches the lower end of the incline?

A rectangular block of ice with dimensions 2 m by 2 m by 0.3 m floats on water. A person weighing 835 N wants to stand on the ice. Would the ice sink below the surface of the water?

1. A room on the lower level of a cruise ship has a 40-cm-diameter circular window. If the midpoint of the window is 2 m below the water surface, determine the hydrostatic force acting on the window, and the pressure center. Take the specific gravity of seawater to be 1.025.

2. The water side of the wall of a 70-m-long dam is a quarter circle with a radius of 7 m. Determine the hydrostatic force on the dam and its line of action when the dam is filled to the rim.

1. Consider an 8-m-long, 8-m-wide, and 2-m-high aboveground swimming pool that is filled with water to the rim.

(a) Determine the hydrostatic force on each wall and the distance of the line of action of this force from the ground.

(b) If the height of the walls of the pool is doubled and the pool is filled, will the hydrostatic force on each wall double or quadruple? Why?

2. Consider a heavy car submerged in water in a lake with a flat bottom. The driver's side door of the car is 1.1 m high and 0.9 m wide, and the top edge of the door is 10 m below the water surface. Determine the net force acting on the door (normal to its surface) and the location of the pressure center if (a) the car is well-sealed, and it contains air at atmospheric pressure and (b) the car is filled with water.

1. The variation of pressure P in gas with density p is given by P = Cpⁿ where C and n and are constants with P = P_o and p = p₀, at elevation z = 0. Obtain a relation for the variation of P with elevation in terms of z. g. n. P_o and p_o.

2. A manometer containing oil (p₀ = 850 kg/m³) is attached to a tank filled with air. If the oil-level difference between the two columns is I50 cm and the atmospheric pressure is 98 kPa, determine the absolute pressure of the air in the tank. 

1. A manometer containing oil (p = 850 kg/m³) is attached to a tank filled with air. If the oil-level difference between the two columns is I50 cm and the atmospheric pressure is 98 kPa, determine the absolute pressure of the air in the tank.

2. Pressure is often given in terms of a liquid column and is expressed as “pressure head." Express the standard atmospheric pressure in terms of (a) mercury (SG = 13.6). (b) water (SG = 1.0). and (c) glycerin (SG = 1.26) columns. Explain why we usually use mercury in manometers.

1. A pressure gauge connected to a tank reads 500 kPa at a location where the atmospheric pressure is 94 kPa. Determine the absolute pressure in the tank.

2. Determine the atmospheric pressure at a location where the barometric reading is 735 mmHg. Take the density of mercury to be 13,600 kg/m³.

1. The 5-m-wide wall is in the form of a parabola. If the depth of the water is h = 4 m, determine the magnitude and direction of the resultant force on the wall.

2. The Tainter gate for a water channel is 1.5 m wide and in the closed position, as shown. Determine the magnitude of the resultant force of the water acting on the gate. Also, what is the smallest torque T that must be applied to open the gate if its weight is 30 kN and its center of gravity is at G.

1. Determine the placement d of the pin on the 0.5-m-wide rectangular gate so that it is on the verge to rotate clockwise (open) when wastewater reaches a height h = 2.5 m. What is the resultant force acting on the gate?

2. The tank is filled to its top with an industrial solvent, ethyl ether. Determine the resultant force acting on the plate ABC, and its location on the plate measured from edge AB of the tank. Use the formula method. Take p_cc = 715 kg/m³.

Part 1. The tide gate opens automatically when the tidewater at B subsides, allowing the marsh at A to drain. For the water level h = 4 m, determine the horizontal reaction at the smooth stop C. The gate has a width of 2 m. At what height h will the gate be on the verge of opening?

Part 2. Determine the critical height h of the water level where the concrete gravity darn starts to tip over. The density of concrete is p_c = 240 Mg/m³. Hint: Work the problem using a 1-m width of the dam.