?A block with mass m = 2.00 kg is placed against a spring on a frictionless incline with

(a) In Problem 8, using energy techniques rather than the techniques of Chapter 4, find the speed of the snowball as it reaches the ground below the cliff. What is that speed (b) if the launch angle is changed to \(41.0^{\circ}\) below the horizontal and (c) if the mass is changed to 2.50 kg? Text Transcription: 41.0^circ

A 5.0 g marble is fired vertically upward using a spring gun. The spring must be compressed 8.0 cm if the marble is to just reach a target 20 m above the marble’s position on the compressed spring. (a) What is the change \(\Delta U_{g}\) in the gravitational potential energy of the marble–Earth system during the 20 m ascent? (b) What is the change \(\Delta U_{s}\) in the elastic potential energy of the spring during its launch of the marble? (c) What is the spring constant of the spring? Text Transcription: Delta U_g Delta U_s

In Fig. 8-35, a runaway truck with failed brakes is moving downgrade at 130 km/h just before the driver steers the truck up a frictionless emergency escape ramp with an inclination of \(\theta=15^{\circ}\). The truck’s mass is \(1.2 \times 10^{4} \mathrm{kg}\). (a) What minimum length L must the ramp have if the truck is to stop (momentarily) along it? (Assume the truck is a particle, and justify that assumption.) Does the minimum length L increase, decrease, or remain the same if (b) the truck’s mass is decreased and (c) its speed is decreased? Text Transcription: theta=15^circ 1.2 times 10^4 kg

A 700 g block is released from rest at height h0 above a vertical spring with spring constant k = 400 N/m and negligible mass. The block sticks to the spring and momentarily stops after compressing the spring 19.0 cm. How much work is done (a) by the block on the spring and (b) by the spring on the block? (c) What is the value of \(h_{0}\)? (d) If the block were released from height \(2.00 h_{0}\) above the spring, what would be the maximum compression of the spring? Text Transcription: h_0 2.00h_0

Figure 8-36 shows an 8.00 kg stone at rest on a spring. The spring is compressed 10.0 cm by the stone. (a) What is the spring constant? (b) The stone is pushed down an additional 30.0 cm and released. What is the elastic potential energy of the compressed spring just before that release? (c) What is the change in the gravitational potential energy of the stone–Earth system when the stone moves from the release point to its maximum height? (d) What is that maximum height, measured from the release point?

A pendulum consists of a 2.0 kg stone swinging on a 4.0 m string of negligible mass. The stone has a speed of 8.0 m/s when it passes its lowest point. (a) What is the speed when the string is at \(60^{\circ}\) to the vertical? (b) What is the greatest angle with the vertical that the string will reach during the stone’s motion? (c) If the potential energy of the pendulum– Earth system is taken to be zero at the stone’s lowest point, what is the total mechanical energy of the system? Text Transcription: 60^circ

Figure 8-34 shows a pendulum of length L = 1.25 m. Its bob (which effectively has all the mass) has speed \(v_{0}\) when the cord makes an angle \(\theta_{0}=40.0^{\circ}\) with the vertical. (a) What is the speed of the bob when it is in its lowest position if \(v_{0}=8.00 \mathrm{~m} / \mathrm{s}\)? What is the least value that \(v_{0}\) can have if the pendulum is to swing down and then up (b) to a horizontal position, and (c) to a vertical position with the cord remaining straight? (d) Do the answers to (b) and (c) increase, decrease, or remain the same if \(\theta_{0}\) is increased by a few degrees? Text Transcription: v_0 theta_0 =40.0^circ v_0 =8.00m/s v_0 theta_0

A 60 kg skier starts from rest at height H = 20 m above the end of a ski-jump ramp (Fig. 8-37) and leaves the ramp at angle \(\theta=28^{\circ}\). Neglect the effects of air resistance and assume the ramp is frictionless. (a) What is the maximum height h of his jump above the end of the ramp? (b) If he increased his weight by putting on a backpack, would h then be greater, less, or the same? Text Transcription: theta=28^circ

The string in Fig. 8-38 is L = 120 cm long, has a ball attached to one end, and is fixed at its other end. The distance d from the fixed end to a fixed peg at point P is 75.0 cm. When the initially stationary ball is released with the string horizontal as shown, it will swing along the dashed arc. What is its speed when it reaches (a) its lowest point and (b) its highest point after the string catches on the peg?

A block of mass m = 2.0 kg is dropped from height h = 40 cm onto a spring of spring constant k = 1960 N/m (Fig. 8-39). Find the maximum distance the spring is compressed.

At t = 0 a 1.0 kg ball is thrown from a tall tower with \(\vec{v}=(18 \mathrm{m} / \mathrm{s}) \hat{\mathrm{i}}+(24 \mathrm{m} / \mathrm{s}) \hat{\mathrm{j}}\). What is \(\Delta U\) of the ball–Earth system between t = 0 and t = 6.0 s (still free fall)? Text Transcription: vec v =(18m/s) hat i +(24m/s) hat j Delta U

A conservative force \(\vec{F}=(6.0 x-12) \hat{i} \mathrm{N}\), where x is in meters, acts on a particle moving along an x axis. The potential energy U associated with this force is assigned a value of 27 J at x = 0. (a) Write an expression for U as a function of x, with U in joules and x in meters. (b) What is the maximum positive potential energy? At what (c) negative value and (d) positive value of x is the potential energy equal to zero? Text Transcription: vec F =(6.0x-12) hat i N

Tarzan, who weighs 688 N, swings from a cliff at the end of a vine 18 m long (Fig. 8-40). From the top of the cliff to the bottom of the swing, he descends by 3.2 m. The vine will break if the force on it exceeds 950 N. (a) Does the vine break? (b) If no, what is the greatest force on it during the swing? If yes, at what angle with the vertical does it break?

Figure 8-41a applies to the spring in a cork gun (Fig. 8?41b); it shows the spring force as a function of the stretch or compression of the spring. The spring is compressed by 5.5 cm and used to propel a 3.8 g cork from the gun. (a) What is the speed of the cork if it is released as the spring passes through its relaxed position? (b) Suppose, instead, that the cork sticks to the spring and stretches it 1.5 cm before separation occurs. What now is the speed of the cork at the time of release?

In Fig. 8-42, a block of mass m = 12 kg is released from rest on a frictionless incline of angle \(\theta=30^{\circ}\). Below the block is a spring that can be compressed 2.0 cm by a force of 270 N. The block momentarily stops when it compresses the spring by 5.5 cm. (a) How far does the block move down the incline from its rest position to this stopping point? (b) What is the speed of the block just as it touches the spring? Text Transcription: theta=30^circ

A 2.0 kg breadbox on a frictionless incline of angle \(\theta=40^{\circ}\) is connected, by a cord that runs over a pulley, to a light spring of spring constant k = 120 N/m, as shown in Fig. 8-43. The box is released from rest when the spring is unstretched. Assume that the pulley is massless and frictionless. (a) What is the speed of the box when it has moved 10 cm down the incline? (b) How far down the incline from its point of release does the box slide before momentarily stopping, and what are the (c) magnitude and (d) direction (up or down the incline) of the box’s acceleration at the instant the box momentarily stops? Text Transcription: theta=40^circ

A block with mass m = 2.00 kg is placed against a spring on a frictionless incline with angle \(\boldsymbol{\theta}=30.0^{\circ}\) (Fig. 8-44). (The block is not attached to the spring.) The spring, with spring constant k = 19.6 N/cm, is compressed 20.0 cm and then released. (a) What is the elastic potential energy of the compressed spring? (b) What is the change in the gravitational potential energy of the block–Earth system as the block moves from the release point to its highest point on the incline? (c) How far along the incline is the highest point from the release point? Text Transcription: theta =30.0^circ

In Fig. 8-45, a chain is held on a frictionless table with one-fourth of its length hanging over the edge. If the chain has length L = 28 cm and mass m = 0.012 kg, how much work is required to pull the hanging part back onto the table?

In Fig. 8-46, a spring with k = 170 N/m is at the top of a frictionless incline of angle \(\theta=37.0^{\circ}\). 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? Text Transcription: theta=37.0^circ

A boy is initially seated on the top of a hemispherical ice mound of radius R = 13.8 m. He begins to slide down the ice, with a negligible initial speed (Fig. 8-47). Approximate the ice as being frictionless. At what height does the boy lose contact with the ice?

In Fig. 8-42, a block of mass m = 3.20 kg slides from rest a distance d down a frictionless incline at angle \(\theta=30.0^{\circ}\) where it runs into a spring of spring constant 431 N/m. When the block momentarily stops, it has compressed the spring by 21.0 cm. What are (a) distance d and (b) the distance between the point of the first block–spring contact and the point where the block’s speed is greatest? Text Transcription: theta=30.0^circ

Two children are playing a game in which they try to hit a small box on the floor with a marble fired from a spring-loaded gun that is mounted on a table. The target box is horizontal distance D = 2.20 m from the edge of the table; see Fig. 8-48. Bobby compresses the spring 1.10 cm, but the center of the marble falls 27.0 cm short of the center of the box. How far should Rhoda compress the spring to score a direct hit? Assume that neither the spring nor the ball encounters friction in the gun.

In Problem 2, what is the speed of the car at (a) point A, (b) point B, and (c) point C? (d) How high will the car go on the last hill, which is too high for it to cross? (e) If we substitute a second car with twice the mass, what then are the answers to (a) through (d)?

(a) In Problem 5, what is the speed of the flake when it reaches the bottom of the bowl? (b) If we substituted a second flake with twice the mass, what would its speed be? (c) If, instead, we gave the flake an initial downward speed along the bowl, would the answer to (a) increase, decrease, or remain the same?

(a) In Problem 4, what initial speed must be given the ball so that it reaches the vertically upward position with zero speed? What then is its speed at (b) the lowest point and (c) the point on the right at which the ball is level with the initial point? (d) If the ball’s mass were doubled, would the answers to (a) through (c) increase, decrease, or remain the same?

In Problem 6, what are the magnitudes of (a) the horizontal component and (b) the vertical component of the net force acting on the block at point Q? (c) At what height h should the block be released from rest so that it is on the verge of losing contact with the track at the top of the loop? (On the verge of losing contact means that the normal force on the block from the track has just then become zero.) (d) Graph the magnitude of the normal force on the block at the top of the loop versus initial height h, for the range h = 0 to h = 6R.

(a) In Problem 7, what is the speed of the ball at the lowest point? (b) Does the speed increase, decrease, or remain the same if the mass is increased?

A uniform cord of length 25 cm and mass 15 g is initially stuck to a ceiling. Later, it hangs vertically from the ceiling with only one end still stuck. What is the change in the gravitational potential energy of the cord with this change in orientation? (Hint: Consider a differential slice of the cord and then use integral calculus.)