Q4-60PP - University Physics with Modern Physics

Question

Forces on a Dancer’s Body. Dancers experience large forces associated with the jumps they make. For example, when a dancer lands after a vertical jump, the force exerted on the head by the neck must exceed the head’s weight by enough to cause the head to slow down and come to rest. The head is about $\mathbf{9}.\mathbf{4}\%$ of a typical person’s mass. Video analysis of a $\mathbf{65}-\mathbf{kg}$ dancer landing after a vertical jump shows that her head decelerates from $\mathbf{4}.\mathbf{0}\mathbf{m}/\mathbf{s}$ to rest in a time of  $\mathbf{0}.\mathbf{20}\mathbf{s}$.

The forces on a dancer can be measured directly when a dancer performs a jump on a force plate that measures the force between her feet and the ground. A graph of force versus time throughout a vertical jump performed on a force plate is shown in Fig. P4.60. What is happening at $\mathbf{0}.\mathbf{4}\mathbf{s}$ ? The dancer is 

(a) bending her legs so that her body is accelerating downward; 

(b) pushing her body up with her legs and is almost ready to leave the ground; (c) in the air and at the top of her jump; 

(d) landing and her feet have just touched the ground.

Step-by-Step Solution

Answer

The dancer is (a) bending her legs so that her body is accelerating downward.

Step 1: Identification of the given data

The given data can be listed below as:

• The mass of the dancer is $m=65\text{\hspace{0.17em}kg}$.
• The mass of the dancer is $9.4\%$ of the mass of the dancer.
• The velocity of her head is $v=4.0\text{\hspace{0.17em}m/s}$.
• The time taken by the dancer to reach the velocity is $t=0.20\text{\hspace{0.17em}s}$.

Step 2: Significance of the force exerted

The force exerted is directly proportional to the mass and the acceleration of the dancer. The force is also directly proportional to the velocity of the dancer.

Step 3: Determination of the movement of the dancer

The weight of the dancer is the initial force of the dancer. According to the graph, it can be identified that the force of the dancer significantly decreases as the dancer bends her leg while preparing for the jump. The force also increases to the peak when the dancer jumps and also drops to the point when she is outside the plate. The force again increases when she again lands on the plate.

 

Thus, the dancer is (a) bending her legs so that her body is accelerating downward.