A block is initially at rest on top of an inclined ramp that makes an angle \( \theta_0 \) with the horizontal. The distance measured along the base of the ramp is \( D \). After the block is released from rest, it slides down the frictionless ramp and then continues onto a rough horizontal surface until it finally comes to rest at the position \( x = 4D \) measured from the base of the ramp. The coefficient of kinetic friction between the block and the rough horizontal surface is \( \mu_k \).(a) On the axes provided, sketch and label graphs of the following quantities as a function of the position \( x \) of the block for \( -D \le x \le 4D \). Both graphs must use the same vertical scale. i. The kinetic energy \( K \) of the block ii. The gravitational potential energy \( U_g \) of the block–Earth system(b) The block is now released from the top of a different ramp that still makes the same angle \( \theta_0 \) with the horizontal but whose base length is \( 2D \). A student is asked whether the block’s final horizontal position will now be twice as far (i.e., at \( x = 8D \)) compared with the original situation. The student reasons that, because the new height is twice the original height, the block will have more energy at the base of the new ramp and therefore will slide farther along the horizontal surface until stopping at \( x = 8D \). i. Which aspects of the student’s reasoning, if any, are correct? If none are correct, write “none”. ii. Which aspects of the student’s reasoning, if any, are incorrect? If none are incorrect, write “none”.(c) Derive an equation for the new final position of the block in terms of \( D \).(d) Referring to the mathematical relationships you obtained in part (c): • For any correct aspects identified in part (b)(i), explain how your relationships support the student’s reasoning. • For any incorrect aspects identified in part (b)(ii), explain how your relationships correct the student’s reasoning.

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A \( 25.0 \) \( \text{kg} \) block is initially at rest on a horizontal surface. A horizontal force of \( 75.0 \) \( \text{N} \) is required to set the block in motion, after which a horizontal force of \( 60.0 \) \( \text{N} \) is required to keep the block moving with constant speed.

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Describe two examples in which the force of friction exerted on an object is in the direction of motion of the object.

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A traffic light hangs from a pole as shown in the accompanying diagram. The uniform aluminum pole AB is of length \( 7.20 \) \( \text{m} \) and has a mass of \( 12.0 \) \( \text{kg} \). The mass of the traffic light is \( 21.5 \) \( \text{kg} \). The point C is located \( 3.80 \) \( \text{m} \) vertically above the pivot A. A massless horizontal cable CD is attached at C and connects to the pole at point D, where the pole makes an angle of \( 37^{\circ} \) with the cable.

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When a box is about to slide but hasn’t moved yet, which friction is acting?

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A car is going through a dip in the road whose curvature approximates a circle of radius \( 200 \) \( \text{m} \). At what velocity will the occupants of the car appear to weigh \( 20\% \) more than their normal weight \( (1.2\,W) \)?

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The occupants of a car traveling at a speed of \( 30 \) \( \text{m/s} \) note that on a particular part of a road their apparent weight is \( 15\% \) higher than their weight when driving on a flat road.

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A car is going over the top of a hill whose curvature approximates a circle of radius \( 350 \) \( \text{m} \). At what velocity will the occupants of the car appear to weigh \( 10\% \) less than their normal weight?

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A loop-de-loop roller coaster has a radius of \( 30 \) \( \text{m} \). Determine the apparent weight a \( 500 \) \( \text{N} \) person will feel at the bottom of the loop while traveling at a speed of \( 25 \) \( \text{m/s} \) and at the top of the loop while traveling at a speed of \( 20 \) \( \text{m/s} \).

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When a falling meteoroid is at a distance above the Earth’s surface of \( 3.00 \) times the Earth’s radius, what is its acceleration due to the Earth’s gravitation?

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Are astronauts really “weightless” while in orbit?

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If I weigh \( 741 \) \( \text{N} \) on Earth at a place where \( g = 9.80 \) \( \text{m/s}^2 \) and \( 5320 \) \( \text{N} \) on the surface of another planet, what is the acceleration due to gravity on that planet?

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Two balls have their centers \( 2.0 \) \( \text{m} \) apart. One ball has a mass of \( 8.0 \) \( \text{kg} \). The other has a mass of \( 6.0 \) \( \text{kg} \). What is the gravitational force between them?

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Two spherical objects have equal masses and experience a gravitational force of \( 25 \) \( \text{N} \) towards one another. Their centers are \( 36 \) \( \text{cm} \) apart. Determine each of their masses.

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An object weighs \( 432 \) \( \text{N} \) on the surface of Earth. At a height of \( 3R_{\text{Earth}} \) above Earth\’s surface, what is its weight?

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Two objects are attracted to each other by a gravitational force \( F \). If each mass is tripled, so that each becomes \( 3 \) times its original value, and the distance between the objects is cut in half to \( \dfrac{1}{2} \) of its original separation, what is the new gravitational force between the objects in terms of \( F \)?

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Determine the distance from the Earth’s center to a point outside the Earth where the gravitational acceleration due to the Earth is \( \frac{1}{10} \) of its value at the Earth’s surface.

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A hypothetical planet has a radius \( 2.0 \) times that of Earth, but has the same mass. What is the acceleration due to gravity near its surface?

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Two objects attract each other gravitationally with a force of \( 2.5 \times 10^{-10} \) \( \text{N} \) when they are \( 0.25 \) \( \text{m} \) apart. Their total mass is \( 4.00 \) \( \text{kg} \). Find their individual masses.

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At what distance from the Earth will a spacecraft traveling directly from the Earth to the Moon experience zero net force because the Earth and Moon pull in opposite directions with equal force?

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Two blocks, A and B, are connected by a light string that passes over a frictionless pulley. Block A, of mass \( 10 \) \( \text{kg} \), rests on a rough plane that makes an angle of \( 45^{\circ} \) with the horizontal, while block B, of mass \( 17 \) \( \text{kg} \), hangs vertically. Starting from rest, what is the minimum coefficient of static friction between block A and the plane required to keep the system in static equilibrium?

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A \( 1 \) \( \text{kg} \) mass on a \( 37^{\circ} \) incline is connected to a \( 3.0 \) \( \text{kg} \) mass on a horizontal surface, as shown. The surfaces and the pulley are frictionless. If \( F = 12 \) \( \text{N} \), (a) what is the acceleration of the system, and (b) what is the tension in the connecting cord?

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A box having a mass of \( 1.5 \) \( \text{kg} \) is accelerated across a table at \( 1.5 \) \( \text{m/s}^2 \). The coefficient of kinetic friction on the box is \( 0.3 \).

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A cart with a mass of \( 20 \) \( \text{kg} \) is pressed against a wall by a horizontal spring with spring constant \( k = 244 \) \( \text{N/m} \) placed between the cart and the wall. The spring is compressed by \( 0.1 \) \( \text{m} \). While the spring is compressed, an additional constant horizontal force of \( 20 \) \( \text{N} \) continues to push the cart toward the wall. What is the resulting acceleration of the cart?

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If the coefficient of static friction is \( \mu_s = 0.5 \), how much force must be applied to a spring (spring constant of \( 0.8 \) \( \text{N/m} \)) which is attached to a block of wood (mass \( 4.0 \) \( \text{kg} \)) in order to just begin to move the block?

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A spring with spring constant \( k = 2.3 \) \( \text{N/m} \) is attached to an object of mass \( 10 \) \( \text{kg} \). If the object is hung from the ceiling by this spring, how much would the spring be stretched?

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A \( 10 \) \( \text{lb} \) force is pushing the \( 40 \) \( \text{lb} \) block shown. The incline angle of the plane is \( \alpha = 40^{\circ} \). The coefficient of static friction between the block and the incline is \( \mu_s = 0.75 \) and the coefficient of kinetic friction is \( \mu_k = 0.65 \). Will the block slide on the plane? If it does, will it slide up or down the plane? What is the friction force between the block and the plane?

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A \( 200 \)\( \text{ lb} \) block is resting on a \( 30^{\circ} \) incline. The coefficient of static friction between the block and the plane is \( \mu_s = 0.8 \). Will the block remain at rest?

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By pressing a painting of mass \( 2.00 \) \( \text{kg} \) against a wall, a man is trying to determine whether it is appropriately positioned. The wall is perpendicular to the pushing force. The coefficient of static friction between the image and the wall is \( 0.660 \). What is the bare minimum pushing force that must be applied?

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A woman stands on a bathroom scale in a motionless elevator. When the elevator begins to move, the scale briefly reads only \( 0.75 \) of her regular weight. Calculate the acceleration of the elevator, and find the direction of acceleration.

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What force is necessary to stretch an ideal spring with a spring constant of \( 120 \) \( \text{N/m} \) by \( 30 \) \( \text{cm} \)?

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A spring with a spring constant of \( 600. \) \( \text{N/m} \) is used for a scale to weigh fish. What is the mass of a fish that would stretch the spring by \( 7.5 \) \( \text{cm} \) from its normal length?

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A spring in a pogo-stick is compressed \( 12 \) \( \text{cm} \) when a \( 40. \) \( \text{kg} \) girl stands on it. What is the spring constant for the pogo-stick spring?

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A spring is connected to a wall and a horizontal force of \( 80.0 \) \( \text{N} \) is applied. It stretches \( 25 \) \( \text{cm} \); what is its spring constant?

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A spring stretches \( 8.0 \) \( \text{cm} \) when a \( 13 \) \( \text{N} \) force is applied. How far does it stretch when a \( 26 \) \( \text{N} \) force is applied?

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A \( 7.3 \) \( \text{kg} \) mass is placed on a spring with a spring constant of \( 34 \) \( \text{N/cm} \). How much does this stretch the spring?

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A spring with a spring constant of \( 50. \) \( \text{N/m} \) is hanging from a stand. A second spring with a spring constant of \( 100. \) \( \text{N/m} \) is hanging from the first spring. How far do they stretch if a \( 0.50 \) \( \text{kg} \) mass is hung from the bottom spring?

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An elastic cord is \( 80. \) \( \text{cm} \) long when it is supporting a mass of \( 10. \) \( \text{kg} \) hanging from it at rest. When an additional \( 4.0 \) \( \text{kg} \) is added, the cord is \( 82.5 \) \( \text{cm} \) long. 

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The figure above shows a cart of mass \( M \) accelerating to the right with acceleration \( a \). A block of mass \( m \) is pressed against the cart’s front vertical surface and is held there only by friction. The coefficient of friction between the block and the cart is \( \mu \). What is the minimum acceleration \( a \) of the cart such that the block will not fall?

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Find the tension in each cable supporting the burglar who weighs \( 600 \) \( \text{N} \). The burglar is at rest, holding a junction point where two cables are attached: one cable is horizontal, and the second cable is attached to the ceiling making an angle of \( 37^{\circ} \) above the horizontal, as shown in the diagram.

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Two wires support an unknown mass as shown in the diagram. The tension in the left wire is measured to be \( 17.5 \) \( \text{N} \) and the tension in the right wire is \( 30.3 \) \( \text{N} \). The left wire makes an angle of \( 30^{\circ} \) with the vertical, and the right wire makes an angle of \( 60^{\circ} \) with the vertical. What is the mass of the object?

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A hockey puck glides on perfectly frictionless ice at constant velocity. Which statement is true?

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A cannon fires a cannonball forward. The recoil of the cannon is backward. Why doesn’t the cannon move backward as fast as the cannonball moves forward?

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A rocket launches upward by expelling exhaust gases downward. This is an illustration of Newton’s ____ Law.

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Two ice skaters push off against each other. Skater A has twice the mass of Skater B. Which statement is correct?

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A car suddenly stops and a passenger lurches forward. This motion is best explained by Newton’s ____ Law.

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During lunch, Alex and Jordan argue about inertia. Alex says if she spins a basketball faster, it will have greater inertia. Jordan argues that inertia only depends on the ball’s mass, not its speed. Who is correct?

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A student kicks a soccer ball. The ball exerts a force back on the student’s foot. Why doesn’t the student’s foot accelerate backward as much as the ball accelerates forward?

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A sled glides across ice and eventually stops. This stopping is best explained by ____.

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A westward–moving car is changing its speed. The net force on the car ____.

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