Chapter 5: Circular Motion; Gravitation

If the acceleration of the particle is always perpendicular to its velocity, it means that the particle is constantly changing its direction but maintaining the same speed. This type of motion is characteristic of circular motion, where the acceleration is always directed towards the center of the circle. Therefore, the particle is moving in a circle.

Centripetal force is the force required to make an object move in a circle. It is directed towards the center of the circle and keeps the object constantly changing direction. Without this force, the object would move in a straight line instead of a curved path. Kinetic friction and static friction are forces that oppose motion and are not specifically related to circular motion. Weight is the force exerted by gravity on an object and is not directly involved in making an object move in a circle.

An object moving with constant speed in a circular path experiences centripetal acceleration. This type of acceleration is directed towards the center of the circle and is necessary to keep the object moving in a curved path. It is caused by the inward force, known as centripetal force, acting on the object. This acceleration does not change the speed of the object, but it does change its direction, allowing it to continuously move in a circular path.

When a car goes around a curve, it experiences a centripetal force that acts towards the center of the curve. This force is necessary to keep the car moving in a curved path and prevent it from moving in a straight line. Therefore, the direction of the net force on the car is towards the curve's center.

As a rocket moves away from the Earth's surface, the rocket's weight decreases. This is because weight is the force exerted on an object due to gravity, and the force of gravity decreases as the distance between the rocket and the Earth's surface increases. Therefore, as the rocket moves further away, the gravitational force acting on it decreases, resulting in a decrease in its weight.

The correct answer is "It is never beyond the pull of Earth's gravity." This is because gravity is a force that extends infinitely into space and is always present, no matter how far away an object is from the source of gravity. Therefore, even when the spaceship is closer to the Moon than to Earth, it is still under the influence of Earth's gravity.

Johannes Kepler was the first person to realize that the planets move in elliptical paths around the Sun. He made this discovery based on the detailed observations and data collected by his mentor, Tycho Brahe. Kepler's laws of planetary motion revolutionized our understanding of the solar system and laid the foundation for modern astronomy.

At the top of the loop, the car must experience a centripetal acceleration towards the center of the loop in order to maintain contact with the track. Since the car is at the top of the loop, the acceleration must be directed downwards, opposite to the direction of gravity. Therefore, the minimum value for the centripetal acceleration at this point is equal to the acceleration due to gravity (g) but in the opposite direction, hence "g downward".

When an object experiences uniform circular motion, it is constantly changing its direction. In order to change its direction, there must be a force acting on the object. This force is directed towards the center of the circular path, as it is responsible for keeping the object moving in a circular path. This force is called the centripetal force. Therefore, the net force in uniform circular motion is directed toward the center of the circular path.

The centripetal acceleration of the satellite would be equal to the acceleration due to gravity (g). This is because centripetal acceleration is the acceleration towards the center of the circular path, and in this case, the gravitational force acts as the centripetal force. Therefore, the centripetal acceleration would be equal to the acceleration due to gravity.

If the particle's acceleration is always perpendicular to its velocity, it means that the velocity vector and acceleration vector are always at right angles to each other. In circular motion, the acceleration vector is always directed towards the center of the circle, while the velocity vector is tangent to the circle. Since the acceleration vector is always perpendicular to the velocity vector in this scenario, it suggests that the particle is moving in a circle.

The centripetal force acting on an object moving in a circular path is given by the equation F = (mv^2)/r, where F is the force, m is the mass, v is the velocity, and r is the radius of the curve. In this case, the centripetal force provided by friction is given as 1.2 * 10^4 N, the radius is 50 m, and the speed is 20 m/s. Rearranging the equation, we can solve for the mass of the car. Therefore, the mass of the car is 1500 kg.

The mass of an object remains the same regardless of its location. Mass is a measure of the amount of matter in an object and is a fundamental property of an object. Therefore, the mass of an object on the Moon is the same as its mass on Earth.

In circular motion, the velocity vector is always tangent to the circular path, while the acceleration vector is directed towards the center of the circle. Since the velocity vector is tangent and the acceleration vector is directed towards the center, they are at right angles to each other, making them perpendicular.

When a car goes around a banked curve, the normal force from the road can be divided into two components: vertical and horizontal. The vertical component of the normal force counteracts the gravitational force acting on the car, which is equal to mg. The horizontal component of the normal force provides the centripetal force required to keep the car moving in a circular path. According to Newton's second law, the centripetal force is given by the equation F = mv^2/r, where m is the mass of the car, v is the speed, and r is the radius of the curve. Therefore, the horizontal component of the normal force must be equal in magnitude to mv^2/r for the car to negotiate the curve successfully.

The gravitational force between two objects is inversely proportional to the square of the distance between the two objects. This means that as the distance between the objects increases, the gravitational force decreases. Conversely, as the distance decreases, the gravitational force increases. This relationship is described by Newton's law of universal gravitation, which states that the force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

Although the speed of an object may be constant, it is still possible for the object to accelerate if the direction of its velocity is changing. Acceleration is defined as any change in velocity, which includes changes in direction. Therefore, even if the speed remains constant, if the object is changing its direction of motion, it is experiencing acceleration.

The quantity that is constant for all planets orbiting the Sun is T^2/R^3. This is because Kepler's third law states that the square of the orbital period of a planet is directly proportional to the cube of its average orbital radius. Therefore, regardless of the specific values of T and R for each planet, their ratio T^2/R^3 will always be constant.

The maximum tension that the string can withstand is 350 N. In order for the string not to break, the centripetal force acting on the mass must not exceed this tension. The centripetal force is given by the equation Fc = m * v^2 / r, where m is the mass, v is the velocity, and r is the radius of the circular path. Rearranging the equation to solve for v, we get v = sqrt(Fc * r / m). Plugging in the given values, we get v = sqrt(350 N * 1.0 m / 0.50 kg) = 26 m/s. Therefore, the maximum speed of the mass is 26 m/s.

The acceleration due to gravity at the surface of a celestial body can be calculated using the formula: acceleration due to gravity = gravitational constant * mass of the celestial body / radius of the celestial body squared. In this case, the mass of the Moon is given as 7.4 * 10^22 kg and its mean radius is given as 1.75 * 10^3 km. By plugging these values into the formula, we get an acceleration due to gravity of approximately 1.6 m/s^2.

When a person goes from sea level to an altitude of 5000 m, their weight changes due to the decrease in gravitational force. The weight of an object is given by the formula W = mg, where m is the mass of the object and g is the acceleration due to gravity. At sea level, g is approximately 9.8 m/s^2, while at an altitude of 5000 m, it decreases slightly. The decrease in g leads to a decrease in weight. Using the formula, the change in weight can be calculated as follows: ΔW = mg(1-altitude/mean radius of the Earth). Plugging in the values, we get ΔW = 100 kg * 9.8 m/s^2 * (1 - 5000 m / (6.38 * 10^6 m)), which simplifies to approximately 1.6 N. Therefore, the correct answer is 1.6 N.

A frictionless curve of radius 100 m, banked at an angle of 45°, may be safely negotiated at a speed of 31 m/s. This is because the angle of banking helps to counteract the centrifugal force acting on the vehicle as it goes around the curve. The higher the speed, the greater the centrifugal force, and the steeper the angle of banking needs to be to counteract it. At a speed of 31 m/s, the angle of banking of 45° is sufficient to keep the vehicle safely on the curve without sliding off.

When a person is standing on a scale in an elevator accelerating downward, the reading on the scale will be less than their true weight. This is because the scale measures the normal force exerted by the person on it, which is equal to their true weight minus the force exerted by the elevator's acceleration. As the elevator accelerates downward, the scale will measure a smaller normal force, resulting in a reading that is less than the person's true weight.

Halley's Comet follows an elliptical orbit around the Sun, which means that its distance from the Sun varies throughout its orbit. According to Kepler's laws of planetary motion, a celestial object moves faster when it is closer to the object it is orbiting. Therefore, as Halley's Comet nears the Sun in its orbit, its speed increases.

The correct answer is 28 m/s. When a curve is banked, it means that the surface of the curve is inclined at an angle to the horizontal. In this case, the curve is banked at 45°. When the curve is frictionless, the only force acting on the car is the normal force, which is perpendicular to the surface of the curve. This normal force can be resolved into two components: one perpendicular to the surface and one parallel to the surface. The component perpendicular to the surface provides the necessary centripetal force to keep the car moving in a circular path. The component parallel to the surface does not affect the motion of the car. By analyzing the forces and applying Newton's second law, it can be determined that the safe speed to take the curve without sliding up or down is 28 m/s.

When a pilot is making an outside vertical loop, the centripetal force required to keep the pilot moving in a circular path is provided by the normal force exerted by the seat. At the top of the loop, the pilot is pushing down on his seat with only one-half of his normal weight. This means that the normal force is half of the pilot's weight. Since the centripetal force is equal to the normal force, it is also half of the pilot's weight. The centripetal force can be calculated using the formula F = mv^2/r, where F is the centripetal force, m is the mass of the pilot, v is the speed, and r is the radius of the loop. By rearranging the formula, we can solve for v, which gives us v = sqrt(Fr/m). Since the centripetal force is half of the pilot's weight, we can substitute F = (1/2)mg into the formula. By plugging in the values, we get v = sqrt((1/2)mg * r/m). The mass of the pilot cancels out, leaving us with v = sqrt((1/2)g * r). Plugging in the values for g (acceleration due to gravity) and r (radius of the loop), we can calculate v, which is approximately 125 m/s.

When the distance between the centers of two objects attracting each other gravitationally is cut in half, the gravitational force between them quadruples. This is because the gravitational force is inversely proportional to the square of the distance between the objects. So, when the distance is halved, the force becomes four times stronger.

To make a turn, the car needs a centripetal force directed towards the center of the curve. This force is provided by the friction between the tires and the road. The centripetal force can be calculated using the formula F = (m * v^2) / r, where m is the mass of the car, v is the velocity, and r is the radius of the curve. Rearranging the formula, we get the frictional force F = (m * v^2) / r. The maximum frictional force is given by the formula F = μ * N, where μ is the coefficient of friction and N is the normal force. The normal force is equal to the weight of the car, which is mg. Equating the two formulas for F, we get (m * v^2) / r = μ * mg. Simplifying, we find μ = (v^2) / (g * r). Plugging in the values given in the question, we get μ = (14^2) / (9.8 * 50) = 0.40. Therefore, the minimum coefficient of friction required for the car to make the turn is 0.40.

The minimum radius of the curve can be determined using the centripetal force equation. The centripetal force is provided by the friction force between the tires and the road. The maximum friction force can be calculated by multiplying the coefficient of friction (0.40) with the normal force (equal to the weight of the car, mg). The centripetal force is equal to the maximum friction force, and it is given by (mv^2)/r, where m is the mass of the car, v is the velocity, and r is the radius of the curve. Equating this to the maximum friction force, we can solve for r, which gives (2.5v^2)/g.

As the car goes around a curve of radius 2r at speed 2v, the centripetal force is given by the equation F = (mv^2)/(2r). Since the radius is twice as big and the speed is also twice as big compared to the first curve, the centripetal force will be (m(2v)^2)/(2(2r)) = (4mv^2)/(4r) = (mv^2)/r. Therefore, the centripetal force on the car as it goes around the second curve is twice as big compared to the first curve.

Yes, this is possible if the speed is changing. When an object moves in a circular path, it experiences centripetal acceleration towards the center of the circle. However, if the speed of the object is changing, it also experiences tangential acceleration in the direction of the velocity vector. This is because the object's velocity is changing, and acceleration is defined as the rate of change of velocity. Therefore, an object moving around a circular path can have both centripetal and tangential acceleration simultaneously.

When an object experiences uniform circular motion, the direction of the acceleration is directed toward the center of the circular path. This is because the object is constantly changing its direction as it moves in a circle, which means it is constantly accelerating towards the center of the circle. The acceleration is necessary to keep the object moving in a curved path, as a force is required to constantly change the direction of the object's velocity.

The weight of an object is directly proportional to the mass of the planet it is on. In this case, the planet has four times the mass of Earth, so the weight of the jar of peanut butter would also be four times its weight on Earth. However, the weight is also inversely proportional to the square of the radius of the planet. Since the radius is twice that of Earth, the weight would be divided by four. Therefore, the weight of the jar of peanut butter on the surface of this planet would be the same as its weight on Earth, which is 12 N.

The surface gravity of a planet is determined by its mass and radius. Since planet B has twice the mass of planet A but the same surface gravity, it must have a larger radius to compensate for the increased mass. The radius of planet B is therefore 1.41 times the radius of planet A.

At the bottom of the path, the tension in the string is equal to the sum of the stone's weight and the centripetal force required to keep it moving in a circle. Since the tension is 3 times the stone's weight, we can write the equation as T = mg + 3mg = 4mg. The centripetal force is given by F = mv^2/r, where v is the speed of the stone. Equating the centripetal force to the tension, we have 4mg = mv^2/r. Simplifying this equation, we get v^2 = 4gr, which means v = sqrt(4gr) or (2gr)^(1/2). Therefore, the stone's speed at the bottom of the path is given by (2gr)^(1/2).

The maximum speed at which a car can round a curve without slipping is determined by the coefficient of friction between the tires and the road surface. Since the coefficient of friction is assumed to be the same for both curves, the maximum speed at which the car can round the second curve can be determined by using the same principle. The radius of the second curve is four times larger than the first curve, so the maximum speed at which the car can round the second curve would be four times larger as well. Therefore, the car can round the second curve with a speed of 40 m/s.

The mass of an object remains the same regardless of the gravitational acceleration. Therefore, the mass of the object on the Moon is still 72 kg. The acceleration of gravity only affects the weight of an object, not its mass.

The average distance from Jupiter to the Sun is 7.9 * 10^11 m. This can be calculated by multiplying the average distance from Earth to the Sun (1.5 * 10^11 m) by a factor of 5.3. Since Jupiter takes 12 years to orbit the Sun once, its average distance can be estimated by multiplying the Earth's distance by the ratio of the orbital periods, which is approximately 5.3.

The acceleration of an object moving in a circular path is given by the formula a = v^2 / r, where v is the velocity and r is the radius of the circle. In this case, the object is moving with a constant speed of 30 m/s and the radius of the circular track is 150 m. Plugging these values into the formula, we get a = (30^2) / 150 = 6.0 m/s^2. Therefore, the acceleration of the object is 6.0 m/s^2.

The centripetal acceleration of an object moving in a circle is given by the formula a = rω², where a is the centripetal acceleration, r is the radius, and ω is the angular velocity. In this case, the radius is given as 0.200 m and the angular velocity is given as 5.00 rev/s. Converting the angular velocity to radians per second (ω = 2πf), we get ω = 2π(5.00) = 31.4 rad/s. Plugging these values into the formula, we get a = (0.200)(31.4)² = 198 m/s².

Since both satellites are rotating in the same orbit, their speeds will be the same regardless of their masses. Therefore, the speed of satellite B is equal to the speed of satellite A.

The correct answer is (4π^2f^2r)/g. This formula represents the minimum coefficient of static friction between the turntable and the coin to prevent slipping. It is derived from the equation for centripetal force, F = mω^2r, where ω is the angular velocity. By equating this force to the maximum static friction force, F_max = μ_smg, and substituting ω = 2πf, the formula (4π^2f^2r)/g is obtained. This equation shows that the coefficient of static friction is directly proportional to the square of the frequency and the radius, and inversely proportional to the acceleration due to gravity.

The centripetal force on an object moving in a circular path can be calculated using the formula F = mv^2/r, where F is the centripetal force, m is the mass of the object, v is the velocity of the object, and r is the radius of the circular path. In this case, the mass of the motorcycle is given as 250 kg, the radius of the turn is given as 13.7 m, and the velocity is given as 96.5 km/h. First, we need to convert the velocity from km/h to m/s by dividing it by 3.6. Then, we can substitute the values into the formula to find the centripetal force. After calculation, the answer is 1.31 * 10^4 N.

When the pilot executes a vertical dive, the force on the plane is the sum of the gravitational force (mg) and the force exerted by the plane to counteract gravity. As the plane follows a semi-circular arc, the force on the pilot increases due to the centripetal force required to keep the plane in the curved path. At the lowest point of the arc, the force on the pilot is the sum of the gravitational force and the centripetal force, which is greater than mg. Since the force is directed upwards to counteract the downward gravitational force, the correct answer is more than mg, and pointing up.

The mutual attractive force between two objects can be calculated using the formula for gravitational force, which is F = G * (m1 * m2) / r^2. In this case, the objects are the electron and proton, and the force is attractive because they have opposite charges. Plugging in the given values for the masses and distance, we can calculate the force to be 3.6 * 10^(-47) N.

The gravitational force between two objects is proportional to the product of the two objects. This means that as the mass of one object increases or as the mass of both objects increases, the gravitational force between them also increases. However, the distance between the two objects does not directly affect the gravitational force.

The speed of a satellite in a low circular orbit is determined by the balance between the gravitational force pulling it towards the Earth and the centrifugal force pushing it away. In order to maintain a stable orbit, these forces must be equal. The formula for the speed of a satellite in a circular orbit is v = √(GM/r), where G is the gravitational constant, M is the mass of the Earth, and r is the radius of the orbit. Given that the mean radius of the Earth is 6.38 * 10^6 m, the correct answer of 7.9 km/s can be calculated using this formula.

An asteroid orbits the Sun in one-third of a year. Since the average distance from the Earth to the Sun is defined as one "astronomical unit" (AU), we can calculate the asteroid's average distance from the Sun by dividing the Earth-Sun distance by the time it takes for the asteroid to complete one orbit. One-third of a year is approximately 0.33 years. Therefore, the asteroid's average distance from the Sun is 1 AU divided by 0.33 years, which is approximately 0.48 AU.

The acceleration of gravity on an object is directly proportional to its mass. Since the acceleration of gravity on the Moon is one-sixth of that on Earth, it means that the Moon's mass is one-sixth of Earth's mass. The radius of the Moon being one-fourth of Earth's radius does not affect the comparison of their masses. Therefore, the Moon's mass compared to the Earth's is 1/6, which is equivalent to 1/6 * 1/4 = 1/24.

The plane is experiencing an acceleration of 4g when pulling out of the dive. This means that the net force acting on the plane is four times the force of gravity. The force causing the acceleration is the centripetal force, given by F = mv^2/r, where m is the mass of the plane, v is its velocity, and r is the radius of curvature. Since the plane is flying at a constant speed of 600 m/s, the force of gravity acting on it is equal to the centripetal force. Therefore, we can set mg = mv^2/r and solve for r. By substituting the given values, we find that r = 9200 m.