In principle, the motion of a simple pendulum and an object on an ideal spring can both be used to provide the basic time interval or period used in a clock. Which of the two kinds of clocks becomes more inaccurate when carried to the top of a high mountain? Justify your answer.
The period of a pendulum depends upon gravity and that or a wound-spring clock does not. What does that tell you about what the answer might be?
When carried to the top of a high mountain, the clock based on a simple pendulum becomes more inaccurate. This is because the acceleration due to gravity decreases as we move to higher altitudes.
To understand why, let's consider the equation that gives the period of a simple pendulum:
T = 2π√(L/g)
where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.
As we move to higher altitudes, the value of g decreases. This means that the period of the pendulum will increase, making the clock run slower. The decrease in g at higher altitudes will result in the pendulum taking longer to complete each swing.
On the other hand, an object on an ideal spring does not rely on the acceleration due to gravity to determine its period. The period of an ideal spring system is given by:
T = 2π√(m/k)
where T is the period, m is the mass of the object, and k is the spring constant.
Since the period of an ideal spring system does not depend on gravitational acceleration, carrying it to the top of a high mountain will not affect its accuracy. The clock based on an ideal spring will continue to keep time accurately regardless of changes in the acceleration due to gravity.
Therefore, the clock based on a simple pendulum becomes more inaccurate when carried to the top of a high mountain due to the decrease in gravitational acceleration, while the clock based on an ideal spring remains unaffected.
To determine which type of clock becomes more inaccurate when taken to the top of a high mountain, we need to consider the factors that affect the period of the pendulum and the object on the spring.
A simple pendulum's period is influenced by two main factors: the length of the pendulum and the acceleration due to gravity. The period of a pendulum is given by the formula:
T = 2π√(L/g),
where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.
On the other hand, the period of an object on an ideal spring is influenced by two factors: the mass of the object and the stiffness of the spring. The period of an ideal spring system is given by the formula:
T = 2π√(m/k),
where T is the period, m is the mass of the object, and k is the stiffness constant of the spring.
When we take these clocks to the top of a high mountain, the acceleration due to gravity decreases because the gravitational pull is slightly weaker at higher altitudes. Therefore, the period of the pendulum increases as gravity is reduced.
On the other hand, the mass of the object on the spring remains constant, and the stiffness of the spring is also unaffected by the change in altitude. Therefore, the period of the object on the spring remains unchanged.
Based on these considerations, it can be concluded that the simple pendulum clock becomes more inaccurate when taken to the top of a high mountain. As the gravity decreases, it will cause the pendulum to slow down and extend its period, resulting in a less accurate measurement of time compared to the unchanged period of the object on the spring clock.