Answer:
Explanation:
Time can be found by dividing the distance by the speed.
The distance is 45 meters and the speed is 12.5 meters per second.
Divide. Note that the meters, or "m" will cancel each other out.
It will take the dolphin 3.6 seconds to swim a distance of 45 meters are 12.5 meters per second.
1. 0.42 Hz
The frequency of a simple harmonic motion for a spring is given by:
where
k = 7 N/m is the spring constant
m = 1 kg is the mass attached to the spring
Substituting these numbers into the formula, we find
2. 2.38 s
The period of the harmonic motion is equal to the reciprocal of the frequency:
where f = 0.42 Hz is the frequency. Substituting into the formula, we find
3. 0.4 m
The amplitude in a simple harmonic motion corresponds to the maximum displacement of the mass-spring system. In this case, the mass is initially displaced by 0.4 m: this means that during its oscillation later, the displacement cannot be larger than this value (otherwise energy conservation would be violated). Therefore, this represents the maximum displacement of the mass-spring system, so it corresponds to the amplitude.
4. 0.19 m
We can solve this part of the problem by using the law of conservation of energy. In fact:
- When the mass is released from equilibrium position, the compression/stretching of the spring is zero: , so the elastic potential energy is zero, and all the mechanical energy of the system is just equal to the kinetic energy of the mass:
where m = 1 kg and v = 0.5 m/s is the initial velocity of the mass
- When the spring reaches the maximum compression/stretching (x=A=amplitude), the velocity of the system is zero, so the kinetic energy is zero, and all the mechanical energy is just elastic potential energy:
Since the total energy must be conserved, we have:
5. Amplitude of the motion: 0.44 m
We can use again the law of conservation of energy.
- is the initial mechanical energy of the system, with
being the initial displacement of the mass and
being the initial velocity
is the mechanical energy of the system when x=A (maximum displacement)
Equalizing the two expressions, we can solve to find A, the amplitude:
6. Maximum velocity: 1.17 m/s
We can use again the law of conservation of energy.
- is the initial mechanical energy of the system, with
being the initial displacement of the mass and
being the initial velocity
is the mechanical energy of the system when x=0, which is when the system has maximum velocity,
Equalizing the two expressions, we can solve to find , the maximum velocity:
Answer:
a) The linear acceleration of the car is , b) The tires did 7.46 revolutions in 2.50 seconds from rest.
Explanation:
a) A tire experiments a general plane motion, which is the sum of rotation and translation. The linear acceleration experimented by the car corresponds to the linear acceleration at the center of the tire with respect to the point of contact between tire and ground, whose magnitude is described by the following formula measured in meters per square second:
Where:
- Magnitude of the radial acceleration, measured in meters per square second.
- Magnitude of the tangent acceleration, measured in meters per square second.
Let suppose that tire is moving on a horizontal ground, since radius of curvature is too big, then radial acceleration tends to be zero. So that:
Where:
- Angular acceleration, measured in radians per square second.
- Radius of rotation (Radius of a tire), measured in meters.
Given that and
. The linear acceleration experimented by the car is:
The linear acceleration of the car is .
b) Assuming that angular acceleration is constant, the following kinematic equation is used:
Where:
- Final angular position, measured in radians.
- Initial angular position, measured in radians.
- Initial angular speed, measured in radians per second.
- Angular acceleration, measured in radians per square second.
- Time, measured in seconds.
If ,
,
, the final angular position is:
Let convert this outcome into revolutions: (1 revolution is equal to 2π radians)
The tires did 7.46 revolutions in 2.50 seconds from rest.