Question
For a particular nonlinear spring, the relationship between the magnitude of the applied force, F, and the stretch of the spring, x, is given by the equation \(F = kx^{1.5}\). How much energy is stored in the spring when is it stretched a distance \(x_1\)
(A) \(\frac{2k_1^{2.5}}{5}\)
(B) \(\frac{kx_1^{1.5}}{5}\)
(C) \(kx_1^{2.5}\)
(D) \(\frac{1}{2}kx_1^2\)
(E) \(1.5 kx_1^{0.5}\)
▶️Answer/Explanation
Ans: A
For a spring that is not linear (i.e., does not obey Hooke’s law) the energy stored is not \(\frac{1}{2} kx^2\). The magnitude of the energy stored will be equal to the magnitude of the work done to stretch the spring to \(x_1\)/ the steps to calculate the work are shown below.
Question
A 2.0 kg mass is attached to the end of a vertical ideal spring with a spring constant of 800 N/m. The mass is pulled down 10 cm from the equilibrium positive and then released, so that it oscillates. The kinetic energy of the 2.0 kg mass at the equilibrium position is
(A) \(\frac{2}{3} J\)
(B) 2 J
(C) 4 J
(D) 12 J
(E) 40 J
▶️Answer/Explanation
Ans: C
The energy of the oscillating spring-mass system will remain constant. When it is pulled down 10 cm energy will be stored in the spring, and when it passes the equilibrium position, all of the energy will be kinetic energy.
Question
Physics students are checking the constant acceleration equations of kinematics by measuring the velocity of a tennis ball that is dropped and falls 6 meters and then passes through a photogate. The predicted velocity is 20% above the velocity measured by the photogate. Which of the following best describes the cause fo the large percent difference?
(A) The ball changes its shape while falling.
(B) The acceleration of gravity varies as the ball is falling.
(C) Air resistance increases the acceleration of the ball.
(D) The acceleration of the balls varies with the velocity.
(E) The acceleration of gravity changes due to air resistance.
▶️Answer/Explanation
Ans: D
The constant kinematics equations ignore the air resistance that decreases the total mechanical energy of the ball as it falls. The force due to air resistance also increases the faster the ball is going, so the force is increasing with time. This would make the acceleration of the ball begin at 9.8 \(m/s^2\) and then decrease as the ball falls. This eliminates (C). The ball is rigid so it will not change shape when falling, which eliminates (A). The acceleration of gravity will be constant over the 6 meters that the ball falls, so this eliminates (B) and (E). Therefore, (D) is left and is correct because the fall is speeding up at a decreasing rate of acceleration.
Question
A disk is rolling without slipping along the ground and the center of mass is traveling at a constant velocity, as shown above. What direction is the acceleration of the contact point P and the center of mass?
▶️Answer/Explanation
Ans: B
The acceleration of the center of mass is zero because the disk is rolling at a constant velocity. The contact point, P, is instantaneously at rest and then moves upward, which implies the acceleration is upward. Combining these two pieces of information indicates (B) is the correct answer.
Question
An object is launched and follows the dashed path shown above. If air resistance is considered, when is the velocity of the object the greatest and teh acceleration of the object the greatest?
▶️Answer/Explanation
Ans: C
The acceleration of the object will be due to gravity and air resistance. Air resistance is proportional to the velocity of the object. Since air resistance decreases the total mechanical energy of the object, the greatest velocity will occur at the point closest to the initial launch, point A. The greatest acceleration will occur when air resistance and gravity are in the same direction and when the object is traveling the fastest. This is also at point A. Therefore, (C) is correct.