Chapter 6: Work And Energy

The correct answer is "Yes, since the choice of the zero of potential energy is arbitrary." This means that the reference point for measuring potential energy can be chosen arbitrarily, and therefore it is possible for a system to have negative potential energy depending on the chosen reference point. The total energy of the system can still be positive, even if the potential energy is negative.

Both objects will travel the same distance before stopping because the distance traveled is determined by the work done on the object, which is equal to the force applied multiplied by the distance traveled. In this case, both objects experience the same braking force, so the work done on each object is the same. Therefore, both objects will travel the same distance before coming to a stop.

The correct answer is watt/second. This is because watt/second is not a unit of power. Power is measured in watts, which is a unit of energy transfer per unit time. The other options, newton-meter/second and joule/second, are not units of power either. Newton-meter/second is a unit of torque, and joule/second is a unit of energy transfer rate.

The power delivered to an object is calculated by multiplying the force applied to the object by its velocity. In this case, the force applied is 10 N and the velocity is 5.0 m/s. Multiplying these values gives us 50 W, which is the power that must be delivered to the object.

The quantity 1/2 mv^2 represents the kinetic energy of an object. Kinetic energy is the energy possessed by an object due to its motion. It is dependent on the mass of the object (m) and the square of its velocity (v^2). The formula 1/2 mv^2 is derived from the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy. Therefore, the correct answer is the kinetic energy of the object.

When the net work done on an object is negative, it means that the work done on the object is in the opposite direction of its motion. This implies that the object is losing energy, resulting in a decrease in its kinetic energy. Therefore, the correct answer is that the object's kinetic energy decreases.

The kinetic friction force between the object and the horizontal surface is 50.0 N. This force opposes the motion of the object and causes it to slow down. The frictional force is directly proportional to the normal force, which is equal to the weight of the object in this case. Therefore, the frictional force can be calculated using the equation Ff = μ * Fn, where μ is the coefficient of kinetic friction. Since the mass of the object is given as 60.0 kg, we can calculate the normal force as Fn = m * g, where g is the acceleration due to gravity. By rearranging the equation, we can solve for the coefficient of kinetic friction, which turns out to be 50.0 N / (60.0 kg * 9.8 m/s^2) = 0.085. Using the equation Ff = μ * Fn, we can now calculate the force of friction as 0.085 * (60.0 kg * 9.8 m/s^2) = 50.0 N. The force of friction can also be expressed as Ff = m * a, where a is the acceleration of the object. Rearranging the equation, we can solve for the acceleration as a = Ff / m = 50.0 N / 60.0 kg = 0.833 m/s^2. The final speed of the object is 0 m/s since it comes to a stop. Using the equation v^2 = u^2 + 2as, where v is the final speed, u is the initial speed, a is the acceleration, and s is the distance, we can solve for the distance as s = (v^2 - u^2) / (2a) = (0^2 - 25.0 m/s^2) / (2 * 0.833 m/s^2) = 375 m. Therefore, the object will slide a distance of 375 m before coming to a stop.

When an object is lifted vertically, work is done against the force of gravity. The work done is equal to the force applied multiplied by the distance over which the force is applied. In this case, the object weighs 90 N and is lifted vertically 2.0 m. Therefore, the work done in lifting the object is 90 N * 2.0 m = 180 J.

If the net work done on an object is zero, it means that the total amount of work done on the object is equal to zero. This implies that there is no change in the object's kinetic energy. Therefore, the object's kinetic energy remains the same.

When a car is driving up a hill, it possesses both kinetic and potential energy. Kinetic energy is the energy of motion, which the car has as it moves up the hill. Potential energy is the energy that an object possesses due to its position or height above the ground. As the car gains height while driving up the hill, it gains potential energy. Therefore, the energy of a car driving up a hill is both kinetic and potential.

The skier starts from rest at the top of the slope, so the initial velocity is 0 m/s. The skier will experience a change in height of 50 m as they go down the slope. Using the equation v^2 = u^2 + 2as, where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement, we can solve for the final velocity. Since the acceleration due to gravity is constant and equal to 9.8 m/s^2, and the displacement is -50 m (negative because the skier is moving downward), we have v^2 = 0^2 + 2(-9.8)(-50). Solving for v, we get v = 31 m/s.

The work done by the weight of the box can be calculated using the formula: work = force x distance x cos(angle). In this case, the force is the weight of the box, which can be calculated using the formula: weight = mass x gravity. The distance is given as 8.0 m and the angle is 30.0 degrees. Plugging in the values, we can calculate the work done by the weight of the box to be 157 J.

The quantity mgy represents the product of the mass (m), the acceleration due to gravity (g), and the height (y) of the object. This equation is derived from the formula for gravitational potential energy, which is given by mgh. The only difference is that y is used instead of h to represent the height. Therefore, mgy represents the gravitational potential energy of the object.

If you did 3 times the work in one-third the time compared to yesterday, it means that your power output increased. To do 3 times the work in one-third the time, you would need to exert 9 times more power than yesterday.

Both masses m(1) and m(2) are released from the same height and experience the same gravitational force. The speed of an object sliding down an inclined plane depends only on the height from which it is released and not on the angle of the incline. Therefore, both masses will have the same speed at the bottom of their respective planes. The masses and angles of the inclines do not affect the speed.

The average power delivered by the engine can be calculated using the formula: Power = Work/Time. In this case, the work done is equal to the change in kinetic energy, which can be calculated using the formula: Work = (1/2)mv^2, where m is the mass of the car and v is the final velocity. The time is given as 7.0 s. By substituting the values into the formulas, we can find the average power delivered by the engine.

The spring constant can be calculated using Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. The equation is given by F = -kx, where F is the force, k is the spring constant, and x is the displacement. In this case, a 10-kg mass causes the spring to stretch 2.0 cm (or 0.02 m). Using the equation, we can calculate the force exerted by the mass on the spring, which is equal to the force exerted by the spring on the mass. Rearranging the equation, we get k = -F/x. Plugging in the values, we get k = -(10 kg * 9.8 m/s^2) / 0.02 m = -490 N/m. Since the spring constant is always positive, we take the absolute value and get 490 N/m, which is equivalent to 4.9 * 10^2 N/m.

When an object hits a wall and bounces back with half of its original speed, its final kinetic energy is proportional to the square of its final speed. Since the final speed is half of the initial speed, the final kinetic energy will be one-fourth (1/4) of the initial kinetic energy. Therefore, the ratio of the final kinetic energy to the initial kinetic energy is 1/4.

The correct unit of work expressed in SI units is kg m^2/s^2. Work is defined as the product of force and displacement, and the SI unit for force is kg m/s^2 (also known as a Newton). Therefore, when force is multiplied by displacement (in meters), the resulting unit for work is kg m^2/s^2.

The area under the curve on a Force versus position graph represents work. This is because work is defined as the product of force and displacement, and the area under the curve represents the integral of force with respect to displacement. Therefore, calculating the area under the curve gives us the amount of work done.

The quantity 1/2 kx^2 represents the elastic potential energy of the object. Elastic potential energy is the energy stored in a stretched or compressed object, and it is directly proportional to the square of the displacement (x) from the equilibrium position and the spring constant (k). Therefore, the given expression, 1/2 kx^2, accurately represents the elastic potential energy of the object.

The correct answer is that they both slide to the same height. Although the heavy object has greater kinetic energy, this does not affect the height it reaches on the hill. The height reached by an object sliding up a hill is determined by its initial speed and the shape of the hill, not its mass or kinetic energy. Therefore, both objects will reach the same height on the hill regardless of their weights or speeds.

The correct answer is 100/(mg) meters. This can be determined by using the principle of conservation of energy. At the highest point in its trajectory, the projectile's kinetic energy is equal to zero, and all of its initial kinetic energy is converted into potential energy. Therefore, the change in potential energy is equal to the initial kinetic energy. Using the formula for potential energy, which is mgh, where m is the mass, g is the acceleration due to gravity, and h is the vertical height, we can set up the equation 220 J = mgh. Solving for h, we get h = 220/(mg) meters. Similarly, at the highest point, the kinetic energy is 120 J, so we can set up the equation 120 J = mgh and solve for h to get h = 120/(mg) meters. Therefore, the projectile rose to a vertical height of 100/(mg) meters.

A 1.0 hp motor is equivalent to 746 W. Since the motor runs for 1.0 hr, we can calculate the energy used by multiplying the power (746 W) by the time (1.0 hr). This gives us 746 J/hr. However, we need to convert the units to joules, so we multiply by 3600 s/hr to get 2,677,600 J, which can be written in scientific notation as 2.7 * 10^6 J. Therefore, the correct answer is 2.7 * 10^6 J.

When the net work done on an object is positive, it means that the total work done on the object is in the same direction as its motion. This indicates that the object is gaining energy and its kinetic energy is increasing. Therefore, the correct answer is "increases."

When the driver slows down her car, the friction force acts in the opposite direction to the motion of the car. This means that the work done by the friction force is negative. The magnitude of the work done can be calculated using the formula W = Fd, where W is the work done, F is the force, and d is the distance. In this case, the force is the friction force and the distance is the distance traveled while slowing down to a stop. Since the work done is negative, it means that energy is being taken away from the car, which is consistent with the car slowing down. The correct answer of -4.84 * 10^5 J indicates the magnitude of the negative work done by the friction force.

The work done by the push can be calculated using the formula: work = force x distance. In this case, the force is the weight of the box, which is given as 400 N. The distance is the length of the inclined plane, which is 4.0 m. Therefore, the work done is 400 N x 4.0 m = 1600 J. However, since the question asks for the work done by the push, we need to consider only the component of the force that is parallel to the direction of motion. This component can be found using trigonometry, and it is equal to the weight of the box multiplied by the cosine of the angle between the inclined plane and the horizontal. Since the plane rises 2.0 m over a length of 4.0 m, the angle can be found using the inverse tangent function: angle = arctan(2.0/4.0) = 26.57 degrees. Therefore, the work done by the push is 400 N x cos(26.57 degrees) x 4.0 m = 800 J.

The centripetal force acting on an object does not do work on the object because the force and the displacement of the object are perpendicular. In order for work to be done, the force and the displacement must be in the same direction. Since the centripetal force always acts towards the center of the circular motion, while the displacement is tangential to the motion, they are perpendicular to each other and no work is done.

The vertical height of the plane can be determined using the equation of motion for an object sliding down an inclined plane. The equation is given by v^2 = u^2 + 2as, where v is the final velocity (9.80 m/s), u is the initial velocity (0 m/s since the object starts from rest), a is the acceleration (which is equal to the acceleration due to gravity, g), and s is the vertical height of the plane. Rearranging the equation, we get s = (v^2 - u^2) / (2a). Substituting the given values, we find s = (9.80^2 - 0^2) / (2 * 9.8) = 4.90 m. Therefore, the vertical height of the plane is 4.90 m.

Car J moves twice as fast as car K, which means that its velocity is twice that of car K. The kinetic energy of an object is proportional to the square of its velocity. Since car J has twice the velocity of car K, its kinetic energy will be four times greater. However, car J has half the mass of car K. The kinetic energy is also proportional to the mass of the object. Since car J has half the mass of car K, its kinetic energy will be half that of car K. Therefore, the ratio of the kinetic energy of car J to car K is 2 to 1.

The ratio of the brick's mass to the rock's mass is 25 to 9 because kinetic energy is directly proportional to mass. Since both objects have the same kinetic energy, the ratio of their masses will be the same as the ratio of their speeds squared. The ratio of the speeds squared is (3^2)/(5^2) = 9/25, so the ratio of the masses is 25 to 9.

The quantity Fd/t represents the power supplied to the object by the force. Power is defined as the rate at which work is done or the amount of energy transferred per unit time. In this case, F represents the force applied to the object, d represents the displacement of the object, and t represents the time taken. Therefore, Fd/t represents the power supplied to the object by the force.

When the arrow is shot into the bale of hay, it comes to a stop after penetrating a certain depth. The average stopping force acting on the arrow can be calculated using the equation F = m * a, where F is the force, m is the mass, and a is the acceleration. In this case, we know the mass of the arrow (20 g) and the velocity at which it strikes the hay (60 m/s). To find the acceleration, we can use the equation v^2 = u^2 + 2as, where v is the final velocity (0 m/s), u is the initial velocity (60 m/s), a is the acceleration, and s is the distance penetrated (20 cm = 0.2 m). Rearranging the equation, we get a = (v^2 - u^2) / (2s). Plugging in the values, we find that the acceleration is -9000 m/s^2. Multiplying this by the mass of the arrow, we get the average stopping force of 180 N.

Work is defined as the transfer of energy that occurs when a force is applied to an object and the object is displaced in the direction of the force. If there is no motion, there is no displacement, and therefore no work is done. Work requires both force and displacement to be present in order to be defined and calculated.

If the car had been traveling twice as fast, it would have had twice the kinetic energy. When the brakes are applied, the car's kinetic energy is converted into heat energy through friction, causing the car to skid. Since the car has twice the initial kinetic energy, it would take four times as much energy to bring the car to a stop, resulting in the car skidding four times farther.

As the planet moves closer to the Sun in its elliptical orbit, it experiences a stronger gravitational force. This increased force causes the planet to accelerate, resulting in an increase in its kinetic energy. Conversely, as the planet moves farther away from the Sun, the gravitational force weakens, causing the planet to decelerate and its kinetic energy to decrease. Therefore, the planet's kinetic energy increases when it approaches the Sun and decreases when it moves farther away.

The correct answer is that the total mechanical energy of a system is constant, only if conservative forces act. This means that if the only forces acting on a system are conservative forces (such as gravity or springs), then the total mechanical energy (the sum of kinetic and potential energy) will remain constant over time. Non-conservative forces, such as friction or air resistance, can convert mechanical energy into other forms (such as heat or sound), causing the total mechanical energy of the system to change.

As the acorn falls from the tree, its potential energy is converted into kinetic energy. The potential energy of the acorn decreases as it falls closer to the ground, while its kinetic energy increases due to its motion. Therefore, the correct answer is that the kinetic energy increases and the potential energy decreases.

When a ball drops and loses gravitational potential energy, it gains an equal amount of kinetic energy according to the law of conservation of energy. Since the ball loses 30 J of gravitational potential energy, it gains 30 J of kinetic energy. Therefore, the correct answer is "exactly 30 J" and not "less than 30 J".

When an object is released from rest, its potential energy is converted into kinetic energy as it falls. The potential energy of an object is given by the equation PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height. In this case, the height is the same for both objects, but the mass of the second object is four times greater than the mass of the first object. Therefore, the potential energy of the second object is four times greater than the potential energy of the first object.

When a ball drops and gains kinetic energy, it means that it is converting gravitational potential energy into kinetic energy. Since the ball gains 30 J of kinetic energy, it must have lost more than 30 J of gravitational potential energy.

When you push against a stationary brick wall, the wall does not move. In order for work to be done, there must be a displacement in the direction of the force applied. Since the wall does not move, there is no displacement and therefore no work is done. Therefore, the amount of work remains constant at zero, regardless of how hard you push.

The change in gravitational potential energy can be calculated using the formula: ΔPE = mgh, where ΔPE is the change in gravitational potential energy, m is the mass of the object, g is the acceleration due to gravity, and h is the change in height. In this case, the mass of the object is 15.0 kg, the acceleration due to gravity is 9.8 m/s^2, and the change in height is 13.0 m - 5.0 m = 8.0 m. Plugging these values into the formula, we get ΔPE = 15.0 kg * 9.8 m/s^2 * 8.0 m = 1176 J. Therefore, the correct answer is 1176 J.

The work done by the weight of the book is calculated using the formula: work = force x distance x cos(theta). In this case, the force is the weight of the book, which is 10 N. The distance is 1.0 m. Since the book is being lifted upwards, the angle between the force and displacement is 180 degrees, so cos(theta) = -1. Therefore, the work done is -10 J.

When a cyclist does work at a certain rate, it means that the power being exerted is equal to that rate. In this case, the cyclist is exerting 500 W of power. Power is defined as the force applied multiplied by the velocity at which the force is applied. Therefore, we can rearrange the equation to solve for force. The force exerted by the cyclist's foot can be calculated by dividing the power (500 W) by the velocity (8.0 m/s), which gives us 63 N.

The net work done on the elevator can be calculated using the formula W = F * d, where W is the work done, F is the force applied, and d is the distance traveled. In this case, the force applied is 5500 N and the distance traveled is 50.0 m. Multiplying these values together gives a net work done of 275,000 J or 2.75 * 10^5 J. Therefore, the correct answer is 2.75 * 10^5 J.

The final kinetic energy of the cart can be calculated using the formula K.E. = 1/2 * m * v^2, where m is the mass of the cart and v is the final velocity. In this case, the force applied is causing the cart to accelerate, so we need to find the final velocity first. We can use Newton's second law of motion, F = m * a, where F is the force applied, m is the mass of the cart, and a is the acceleration. Rearranging the equation, we get a = F/m. Plugging in the values, we have a = 200 N / 55 kg = 3.64 m/s^2. Now, we can use the kinematic equation v^2 = u^2 + 2as, where u is the initial velocity (which is 0 m/s since the cart is initially at rest), a is the acceleration, and s is the distance traveled. Plugging in the values, we have v^2 = 0 + 2 * 3.64 m/s^2 * 10 m = 72.8 m^2/s^2. Taking the square root of both sides, we get v = 8.53 m/s. Now, we can calculate the final kinetic energy using the formula K.E. = 1/2 * m * v^2. Plugging in the values, we have K.E. = 1/2 * 55 kg * (8.53 m/s)^2 = 2.0 * 10^3 J. Therefore, the correct answer is 2.0 * 10^3 J.

When an object is held in a stationary position, no work is being done on the object. Work is defined as the transfer of energy that occurs when a force is applied over a distance. In this case, the object is not moving vertically, so there is no displacement and therefore no work is being done. Therefore, the correct answer is 0 J.

When the ball is thrown straight up, the work done by gravity is negative because the force of gravity is acting in the opposite direction of the displacement of the ball. As the ball comes back down, the work done by gravity is positive because the force of gravity is now acting in the same direction as the displacement of the ball. Therefore, the correct answer is that the work is - on the way up and + on the way down.

When the ball is 0.70 m above the floor, its potential energy is given by mgh, where m is the mass of the ball (1.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height above the floor (0.70 m). Its kinetic energy is given by (1/2)mv^2, where v is the velocity of the ball. Since the potential energy equals the kinetic energy, we can equate the two equations and solve for v. By substituting the given values into the equation and solving for v, we find that the ball is moving at a speed of 3.7 m/s.