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Q.7

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Physics for Scientists and Engineers: A Strategic Approach with Modern Physics
Found in: Page 199

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Short Answer

A block on a -long string swings in a circle on a horizontal, frictionless table at .

a. What is the speed of the block?

b. What is the tension in the string?

a). The speed of the block is .

b). The tension in the string is .

See the step by step solution

Step by Step Solution

Step 1:Given Information (Part a)

A block on a -long string swings in a circle on a horizontal, frictionless table at .

Step 2: Explanation (Part a)

The block velocity is tangent to the circle of motion, and its acceleration is called the centripetal acceleration and it points toward the center of the circle. The block has angular velocity and a tangential speed and they are related to each other by

Where is the radius of the circle. The angular velocity is given by . In units, the angular frequency has unit radians per second ( . So, let us convert the unit from and

Now, plug the values for and into equation (1) to get

Step 3: Final Answer (Part a)

The speed of the block is .

Step 4: Given Information (Part b)

A block on a -long string swings in a circle on a horizontal, frictionless table at .

Step 5: Explanation (Part b)

This acceleration of the block is given by equation (8.4) in the form

Where is the radius of the circle and is the velocity of the block that we are calculated in part (a). At a uniform circular motion, the velocity vector has a tangential component and the acceleration vector has a radial component. From Newton 's first law, the block has a net force exerted on it as it doesn't move with a constant velocity. So, using the expression of the acceleration from equation (2), we get the net force by

Step 6: Explanation (Part b)

This force has a direction toward the center of the circular motion. The block moves in a circle with a radius . Now, we plug the values for and into equation (3) to get

This force is an identifiable agent and it is one of our familiar forces such as friction, normal or tension forces. The block moves on the table in a circular motion, there is no friction force because the table is frictionless, also there is no horizontally normal force, so the tension force in the wire causes this force. Therefore, the tension force will be

Step 7: Final Answer (Part b)

The tension force will be .

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