A 300 g block on a 58.0-cm-long string swings in a circle on a horizontal, frictionless table at 95.0 rpm. What is the speed of the block?

To determine the speed of the block swinging in a circle, we'll use the relationship between circular motion and speed.

### Given:
- Mass of the block, \( m = 300 \, \text{g} = 0.300 \, \text{kg} \)
- Length of the string (radius of the circle), \( r = 58.0 \, \text{cm} = 0.580 \, \text{m} \)
- Rotation speed, \( f = 95.0 \, \text{rpm} \)

### Required:
- Speed of the block, \( v \)

### Step 1: Convert rpm to rad/s
The speed of the block depends on the angular velocity, which is related to the frequency (in revolutions per minute, rpm). We need to convert the angular velocity to radians per second (\(\omega\)):

\[
\omega = 2 \pi f
\]

where:
- \( f = \frac{95.0 \, \text{rpm}}{60 \, \text{s/min}} = 1.5833 \, \text{rps} \) (revolutions per second)

\[
\omega = 2 \pi \times 1.5833 \approx 9.948 \, \text{rad/s}
\]

### Step 2: Find the linear speed \( v \)
The linear speed \( v \) is related to the angular speed \( \omega \) and the radius \( r \):

\[
v = \omega r
\]

Substitute the values:

\[
v = 9.948 \, \text{rad/s} \times 0.580 \, \text{m}
\]

\[
v \approx 5.77 \, \text{m/s}
\]

### Final Answer:
The speed of the block is approximately **5.77 m/s**.

Would you like more details on how circular motion equations are derived or any other part of the solution?

Here are some related questions for further practice:
1. What is the centripetal force acting on the block in this problem?
2. How would the speed change if the string length doubled?
3. What would happen to the speed if the mass of the block was increased?
4. How does friction affect the motion if the surface is not frictionless?
5. How would you calculate the tension in the string during circular motion?

**Tip:** Remember, in uniform circular motion, the speed remains constant, but the direction changes continuously, leading to centripetal acceleration.

This problem involves finding the speed of a 300 g block swinging in a circle on a frictionless table using the concepts of angular velocity and linear speed. Key details include the mass of the block, length of the string, and rotation speed of 95 rpm. This example is ideal for high school physics students studying circular motion in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation