To solve this problem, we need to analyze the forces acting on the body and how they relate to its motion. We have a mass of 0.5 kg, a coefficient of friction of 0.2, and a horizontal force applied that varies with time as \( F = 2 \sin(t) \). Our goal is to determine the velocity of the body when its acceleration becomes zero for the first time after it starts moving.
Understanding the Forces at Play
First, let’s identify the forces acting on the mass:
- Applied Force (F): This is given as \( F = 2 \sin(t) \).
- Frictional Force (f): The frictional force can be calculated using the formula \( f = \mu N \), where \( \mu \) is the coefficient of friction and \( N \) is the normal force. Since the mass is on a horizontal surface, \( N = mg \), where \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)). Thus, \( f = 0.2 \times 0.5 \times 9.81 = 0.981 \, \text{N} \).
Setting Up the Equation of Motion
Next, we can set up Newton's second law, which states that the net force acting on the mass is equal to the mass times its acceleration:
Net Force \( F_{\text{net}} = F - f \)
Substituting the values we have:
Net Force \( = 2 \sin(t) - 0.981 \)
According to Newton's second law:
Mass \( \times \) Acceleration \( = 0.5 \times a \)
Thus, we can write:
0.5 \( a = 2 \sin(t) - 0.981 \)
Finding Acceleration and Velocity
Rearranging gives us the acceleration:
Acceleration \( a = \frac{2 \sin(t) - 0.981}{0.5} = 4 \sin(t) - 1.962 \)
To find when the acceleration becomes zero for the first time, we set \( a = 0 \):
0 \( = 4 \sin(t) - 1.962 \
Solving for \( \sin(t) \):
4 \( \sin(t) = 1.962 \)
\( \sin(t) = \frac{1.962}{4} = 0.4905 \)
Now, we can find \( t \) using the inverse sine function:
\( t = \sin^{-1}(0.4905) \approx 0.515 \, \text{s} \)
Calculating the Velocity
To find the velocity at this time, we need to integrate the acceleration over time. The velocity \( v \) can be found by integrating the acceleration function:
Since \( a = 4 \sin(t) - 1.962 \), we can integrate this with respect to \( t \):
\( v(t) = \int (4 \sin(t) - 1.962) \, dt \)
Calculating the integral gives:
\( v(t) = -4 \cos(t) - 1.962t + C \)
To find the constant \( C \), we assume the initial velocity \( v(0) = 0 \):
\( v(0) = -4 \cos(0) - 1.962(0) + C = 0 \)
Thus, \( C = 4 \).
Now, substituting back into the velocity equation:
\( v(t) = -4 \cos(t) - 1.962t + 4 \)
Now we can find \( v(0.515) \):
\( v(0.515) = -4 \cos(0.515) - 1.962(0.515) + 4 \)
Calculating \( \cos(0.515) \approx 0.857 \):
\( v(0.515) = -4(0.857) - 1.962(0.515) + 4 \)
\( v(0.515) \approx -3.428 - 1.012 + 4 \approx -0.440 \, \text{m/s} \)
Since we are interested in the speed, we take the absolute value. However, this indicates that the body is not moving forward at this point, as the friction is greater than the applied force. Thus, the body does not reach a positive velocity before the acceleration becomes zero.
Final Thoughts
Based on the calculations, the velocity of the body when its acceleration becomes zero for the first time is not among the options provided. Therefore, the correct answer is (d) none of these.
