Problem 1
A goalkeeper kicks a ball of mass 400 g in the air, and reaches a position of height 15.9 m above the ground corresponding to a horizontal distance 19.7 m from its initial position. During its flight, the ball experiences a horizontal friction force of 3 N from a wind blowing opposite to the horizontal direction of its motion.
1) What is the work done by the gravitational force on the ball?
2) What is the work done by the friction force on the ball?
3) What is the total work done on the ball?
Problem 2
When a mass of 2 kg is hung vertically at the end of two light springs connected in series, the combined stretching of the two springs is 10 cm .
1) If the spring constant of the first spring is k1 = 294 N/m, what is the spring constant of the second spring k2?
2) If the 2 kg mass is removed, how much work should be done by an applied force to stretch the two springs by 20 cm from their unstretched position (see figure)?
Problem 3
An object of mass 5 kg is thrown up on a rough incline of 20o with an initial speed of 2 m/s, and stops at a distance d at the top of the incline . The coefficient of kinetic friction between the object and ground is k= 0.1
(1) Using the work-kinetic energy theorem, determine the distance d.
(2) What is the average power used?
Problem 1:
1) To find the work done by the gravitational force on the ball, we can use the formula:
Work = Force x Distance
The gravitational force can be calculated using the mass of the ball and the acceleration due to gravity. Since the ball is kicked upwards, the work done by gravity is negative because the force of gravity is acting in the opposite direction to the displacement. Therefore, we can calculate the work done by gravity as:
Work = -(mass x gravity x height)
Substituting the values given, the mass (converted to kilograms) = 0.4 kg, gravity = 9.8 m/s^2, and height = 15.9 m, we can calculate the work done by gravity.
2) To find the work done by the friction force on the ball, we can use the formula:
Work = Force x Distance
Since the friction force acts horizontally, its direction is opposite to the displacement of the ball. The work done by friction is also negative, as the force and displacement are in opposite directions. So we can calculate the work done by friction as:
Work = -(friction force x horizontal distance)
Using the given value of the friction force, which is 3 N, and the horizontal distance of 19.7 m, we can calculate the work done by friction.
3) To find the total work done on the ball, we need to sum up the work done by gravity and the work done by friction, as they are acting in different directions. The total work done is the algebraic sum of these two work values.
Total Work = Work done by gravity + Work done by friction
Substituting the calculated values for the work done by gravity and the work done by friction, we can find the total work done on the ball.
Problem 2:
1) To find the spring constant of the second spring, we can use the formula:
Combined spring constant = (k1 x k2) / (k1 + k2)
Since the combined stretching of the two springs is 10 cm, which is equal to 0.10 m, we can substitute this value along with the given spring constant of the first spring (k1 = 294 N/m) into the formula. Solving for k2 will give us the spring constant of the second spring.
2) To find the work done by an applied force to stretch the two springs by 20 cm, we can use the formula:
Work = (1/2) x k1 x (x1^2) + (1/2) x k2 x (x2^2)
Here, k1 is the spring constant of the first spring, which can be taken as the value obtained in the previous question (or 294 N/m), x1 is the initial displacement of the first spring, x2 is the initial displacement of the second spring (which can be calculated based on the combined stretching and the initial displacements of the individual springs), and k2 is the spring constant of the second spring (obtained from the first question). Substituting the given values, we can calculate the work done by the applied force.
Problem 3:
1) To determine the distance d using the work-kinetic energy theorem, we need to find the work done by all the forces (friction and gravity) on the object. The work-kinetic energy theorem states that the net work done on an object is equal to its change in kinetic energy.
Net Work = Change in Kinetic Energy
Since the object is thrown up on the incline and comes to a stop at the top, its initial kinetic energy is zero. Therefore, the net work done on the object is equal to its initial potential energy at height d.
Net Work = Potential Energy at height d
The potential energy can be calculated using the formula:
Potential Energy = mass x gravity x height
By equating the net work done to the potential energy, we can solve for the distance d.
2) The average power used can be calculated using the formula:
Average Power = Work done / Time
To find the work done, we need to calculate the total work done against friction and against gravity. The work done against gravity can be calculated using the formula:
Work = force x distance
Where the force is equal to the weight of the object and the distance is equal to the height d. The work done against friction can be calculated using the formula:
Work = force x distance
Where the force can be obtained using the coefficient of kinetic friction, the weight of the object, and the angle of the incline. The distance is still the height d. Once the work done is calculated, we can divide it by the time taken to find the average power used.