Question

Conceptual Question 6.14 Playground swings move through an arc of a circle. Part A When you are on a swing, and at the lowest point of your motion, is your apparent weight greater than, less than, or equal to your true weight? Explain Drag the terms on the left to the appropriate blanks on the right to complete the sentences. In circular motion there must be a centripetal force directed the center. At the bottom of the swing trajectory we can pretend you are in uniform circular motion, so the net force points toward the center of the circle. Therefore, the swing must exert a force on you than the earth does in the downward direction. So, your apparent weight is your true weight. toward less than up smaller downward equal to larger upward greater than down from

Answer 1

Concept and reason

The concepts used to solve this problem are force, apparent weight, true weight and circular motion.

When the object is at swing and at the bottom of the swing trajectory, we can predict that the object is in uniform circular motion. To maintain the circular motion, a force acts towards the center of the path. Hence, the object has two forces acting on it. The two forces are, force due to gravity and the centripetal force. At the bottom of the arc, the object requires a maximum force to hold it in its circular path.

Apparent weight defines the heaviness of a given object. When the force of gravity and normal force does not balance each other, the apparent weight of an object will differ from the weight of an object.

Fundamentals

When an object is in uniform circular motion, it will experience a centripetal force towards the center of circular path.

The expression for the centripetal force is,

$F = \frac{{m{v^2}}}{r}$

Here, m is the mass of the object, v is velocity of the object, and r is radius of circular path.

At the lowest point the resultant force is,

$F = N - mg$

Here, N is the apparent weight of the object and g is acceleration due to gravity.

The centripetal force is supplied by the resultant force. Hence, the expression for the apparent weight is,

$N = mg + \frac{{m{v^2}}}{r}$

(A.1)

At the lowest point, the resultant force is,

${F_{{\rm{re}}}} = N - mg$

Here, ${F_{re}}$ is the resultant force.

The expression for the centripetal force is,

$F = \frac{{m{v^2}}}{r}$

The resultant force supplies the centripetal force.

Substitute N – mg for F_re in the above expression.

$\begin{array}{c}\\N - mg = \frac{{m{v^2}}}{r}\\\\N = mg + \frac{{m{v^2}}}{r}\\\end{array}$

Here, mg is the actual weight.

An additional centrifugal force also acts with the actual weight. Therefore the apparent weight is greater than true weight.

(A.2.1)

In the case of circular motion, the centripetal forces keep the particle in the circular path.

In the case of circular motion, even if the particle is at constant speed, its velocity will constantly change due to the change in direction of the object.

Hence, the object must be accelerating and the centripetal force is the resultant force that causes this acceleration. This force will be directed toward the center of the circle.

Therefore, in the circular motion, there must be a centripetal force directed towards the center.

(A.2.2)

At the bottom of the swing trajectory, we can pretend that the object is in uniform circular motion.

In the case of uniform circular motion, the net force will be the centripetal force which keeps the object in the circular path.

The centripetal force points up towards the center of the circle.

Therefore, at the bottom of the swing trajectory, we can pretend that the object is in uniform circular motion, so the net force points up towards the center of the circle.

(A.2.3)

When the object is on a swing and at the bottom of the swing trajectory, two forces will act. One is force due to gravity which acts in a downward direction.

The expression for force due to gravity is,

$F = mg$

The second one is centripetal force along with the force due to gravity which is exerted by swing directed upwards.

The expression for upward force is,

$F = mg + \frac{{m{v^2}}}{r}$

Therefore, the swing must exert a larger force on the object in an upward direction than to the earth in the downward direction.

(A.2.4)

At the lowest point, the resultant force is,

${F_{re}} = N - mg$

The expression for the centripetal force is,

$F = \frac{{m{v^2}}}{r}$

The resultant force supplies the centripetal force.

$\begin{array}{c}\\{F_{re}} = \frac{{m{v^2}}}{r}\\\\N - mg = \frac{{m{v^2}}}{r}\\\\N = mg + \frac{{m{v^2}}}{r}\\\end{array}$

Here, mg is the actual weight.

An additional centrifugal force also acts with the actual weight.

Therefore, the apparent weight is greater than true weight.

Ans: Part A.1

The apparent weight is greater than true weight.

Part A.2.1

In the circular motion, there must be a centripetal force directed towards the center.

Part A.2.2

At the bottom of the swing trajectory, we can pretend that the object is in uniform circular motion, so the net force points up towards the center of the circle.

Part A.2.3

Therefore, the swing must exert a larger force on the object in an upward direction than the earth does in the downward direction.

Part A.2.4

So, your apparent weight is greater than true weight.